[seqfan] Re: reply to Message 4. Anyone recognize this matrix? (Neil Sloane)
Marc LeBrun
mlb at well.com
Sat Jan 1 20:48:07 CET 2022
Just responding to an old thread (no idea if this was subsequently addressed independently):
I would suggest that rather than rearranging an existing sequence, that a new sequence be added, with the new formula -- then both sequences outfitted with appropriate cross-references to each other and to A341915 and A341916, etc.
> On Apr 22, 2021, at 1:25 PM, I.V. Serov <i.v.serov at chf.nu> wrote:
>
> Dear Neil,
>
> Would you allow to rearrange the order of the rows in the matrix?
>
> It looks like the same rows can be generated (in a different order) by means of a formula:
>
> B(1) = 01.
> B(2n+0) = concatenate(B'(n);B(n)).
> B(2n+1) = concatenate(B(n);B(n)').
> Here B(n)' equals B(n) with the rightmost bit flipped.
>
> These are the first 31 rows of your matrix in the new order:
>
> 01*
> 0001*
> 0100*
> 00000001*
> 00010000*
> 01010100*
> 01000101*
> 0000000000000001*
> 0000000100000000*
> 0001000100010000*
> 0001000000010001*
> 0101010101010100*
> 0101010001010101*
> 0100010001000101*
> 0100010101000100*
> 00000000000000000000000000000001*
> 00000000000000010000000000000000*
> 00000001000000010000000100000000*
> 00000001000000000000000100000001*
> 00010001000100010001000100010000*
> 00010001000100000001000100010001*
> 00010000000100000001000000010001*
> 00010000000100010001000000010000*
> 01010101010101010101010101010100*
> 01010101010101000101010101010101*
> 01010100010101000101010001010101*
> 01010100010101010101010001010100*
> 01000100010001000100010001000101*
> 01000100010001010100010001000100*
> 01000101010001010100010101000100*
> 01000101010001000100010101000101*
>
> The order permutations are described by A341915 and A341916.
>
>
> Kind regards,
>
> Igor Serov
> www.chf.nu
>
>
>
> On 21-04-2021 15:07, seqfan-request at list.seqfan.eu wrote:
>> Send SeqFan mailing list submissions to
>> seqfan at list.seqfan.eu
>> To subscribe or unsubscribe via the World Wide Web, visit
>> http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan
>> or, via email, send a message with subject or body 'help' to
>> seqfan-request at list.seqfan.eu
>> You can reach the person managing the list at
>> seqfan-owner at list.seqfan.eu
>> When replying, please edit your Subject line so it is more specific
>> than "Re: Contents of SeqFan digest..."
>> Today's Topics:
>> 1. A339950, A189378 (Don Reble)
>> 2. Re: A339950, A189378 (Jeffrey Shallit)
>> 3. How to define analogue to this sequence pertaining to Roman
>> numerals (Alonso Del Arte)
>> 4. Anyone recognize this matrix? (Neil Sloane)
>> 5. Recent perturbations on the Seqfan Mailing List (Olivier Gerard)
>> 6. Conjecture: a(n) = n only if n = 1 or 9 (Alonso Del Arte)
>> 7. Lucasian (pseudo)primes (Tomasz Ordowski)
>> 8. Re: Conjecture: a(n) = n only if n = 1 or 9 (D. S. McNeil)
>> 9. Sum-Product Problem (Frank Adams-watters)
>> 10. Planar distributive lattices (Allan Wechsler)
>> 11. Re: Sum-Product Problem (Neil Sloane)
>> 12. Re: Sum-Product Problem (Neil Sloane)
>> 13. Re: Planar distributive lattices (Neil Sloane)
>> 14. Re: Planar distributive lattices (Neil Sloane)
>> 15. Re: Sum-Product Problem (Hugo Pfoertner)
>> 16. Re: Conjecture: a(n) = n only if n = 1 or 9 (Alonso Del Arte)
>> 17. Re: Planar distributive lattices (Allan Wechsler)
>> 18. Re: Planar distributive lattices (Neil Sloane)
>> 19. help naming/describing sequences for bounds of Goldbach's
>> Comet, was: what happened to my A342302, its been replaced ???
>> (peter lawrence)
>> 20. Re: Anyone recognize this matrix? (Richard J. Mathar)
>> 21. Re: Anyone recognize this matrix? (Neil Sloane)
>> ----------------------------------------------------------------------
>> Message: 1
>> Date: Tue, 6 Apr 2021 02:19:20 -0600
>> From: Don Reble <djr at nk.ca>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] A339950, A189378
>> Message-ID: <606C1988.90801 at nk.ca>
>> Content-Type: text/plain; charset=UTF-8; format=flowed
>> Seqfans:
>> I computed more of A339950 (1,7,14,20,...,391,397,404).
>> So far, A339950(n+1) = A189378(n)+1.
>> Does anyone see how to (dis)prove that?
>> --
>> Don Reble djr at nk.ca
>> ------------------------------
>> Message: 2
>> Date: Tue, 6 Apr 2021 13:10:45 -0400
>> From: Jeffrey Shallit <shallit at uwaterloo.ca>
>> To: <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: A339950, A189378
>> Message-ID: <9ad00bb5-2f24-5cd5-3014-591013221ab3 at uwaterloo.ca>
>> Content-Type: text/plain; charset="utf-8"; format=flowed
>> Yes, I proved it this morning with the automatic theorem-prover "Walnut".
>> On 2021-04-06 4:19 a.m., Don Reble via SeqFan wrote:
>>> Seqfans:
>>> ? I computed more of A339950 (1,7,14,20,...,391,397,404).
>>> ? So far, A339950(n+1) = A189378(n)+1.
>>> ? Does anyone see how to (dis)prove that?
>> ------------------------------
>> Message: 3
>> Date: Thu, 8 Apr 2021 17:12:24 -0400
>> From: Alonso Del Arte <alonso.delarte at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] How to define analogue to this sequence pertaining
>> to Roman numerals
>> Message-ID:
>> <CAGyGvfVOmJxnkd6s3Zuq5pNvOHDYVV2yeWrNPkYQQgHYBNN28A at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> Almost all integers from 1 to 3999 are Roman numeral Harshad numbers. Much
>> fewer of them can have their Roman numeral representations built up from
>> the Roman numeral representations of their nontrivial divisors. For
>> example, the divisors of 80 include 20 and 40, which in Roman numerals are
>> XX and XL, respectively. From those we can assemble LXXX. I'm not
>> interested in requiring that all the "digits" of a divisor be used, e.g.,
>> for LXXX we could use X, XX and XL, using XL only for the L and discarding
>> the extra X.
>> scala> (1 to 3999).filter(romNumDivAnagrammable)
>> res12: IndexedSeq[Int] = Vector(8, 18, 36, 80, 84, 88, 180, 184, 186, 270,
>> 276, 282, 288, 360, 372, 380, 384, 396, 800, 804, 808, 810, 812, 816, 820,
>> 822, 828, 832, 834, 836, 840, 846, 848, 852, 858, 860, 864, 868, 870, 876,
>> 880, 882, 884, 888, 894, 896, 1800, 1804, 1806, 1808, 1812, 1816, 1818,
>> 1820, 1824, 1830, 1832, 1836, 1840, 1848, 1856, 1860, 1864, 1872, 1876,
>> 1880, 1884, 1888, 1890, 1896, 2700, 2706, 2712, 2718, 2724, 2730, 2736,
>> 2742, 2748, 2754, 2760, 2766, 2772, 2778, 2784, 2790, 2796, 2802, 2808,
>> 2814, 2820, 2826, 2832, 2838, 2844, 2850, 2856, 2862, 2868, 2874, 2880,
>> 2886, 2892, 2898, 3600, 3612, 3624, 3636, 3648, 3660, 3672, 3684, 3696,
>> 3708, 3720, 3732, 3744, 3756, 3768, 3780, 3792, 3800, 3804, 3808, 3810,
>> 3816, 3820, 3822, 3824, 3828, 3832, 3834, 3836, 3840, 3846, 3848, 38...
>> scala> romNums(res21)
>> res23: IndexedSeq[numerics.RomanNumeralsNumber] = Vector(MMMDCCCXXIV,
>> MMMDCCCXXVIII, MMMDCCCXXXII, MMMDCCCXXXIV, MMMDCCCXXXVI, MMMDCCCXL,
>> MMMDCCCXLVI, MMMDCCCXLVIII, MMMDCCCLII, MMMDCCCLVI, MMMDCCCLVIII,
>> MMMDCCCLX, MMMDCCCLXIV, MMMDCCCLXX, MMMDCCCLXXII, MMMDCCCLXXVI,
>> MMMDCCCLXXX, MMMDCCCLXXXII, MMMDCCCLXXXVIII, MMMDCCCXCII, MMMDCCCXCIV,
>> MMMCM, MMMCMXII, MMMCMXXIV, MMMCMXXXVI, MMMCMXLVIII, MMMCMLX, MMMCMLXXII,
>> MMMCMLXXXIV, MMMCMXCVI)
>> scala> romNums(res18)
>> res24: IndexedSeq[numerics.RomanNumeralsNumber] = Vector(VIII, XVIII,
>> XXXVI, LXXX, LXXXIV, LXXXVIII, CLXXX, CLXXXIV, CLXXXVI, CCLXX, CCLXXVI,
>> CCLXXXII, CCLXXXVIII, CCCLX, CCCLXXII, CCCLXXX, CCCLXXXIV, CCCXCVI, DCCC,
>> DCCCIV, DCCCVIII, DCCCX, DCCCXII, DCCCXVI, DCCCXX, DCCCXXII, DCCCXXVIII,
>> DCCCXXXII, DCCCXXXIV, DCCCXXXVI, DCCCXL, DCCCXLVI, DCCCXLVIII, DCCCLII,
>> DCCCLVIII, DCCCLX, DCCCLXIV, DCCCLXVIII, DCCCLXX, DCCCLXXVI, DCCCLXXX,
>> DCCCLXXXII, DCCCLXXXIV, DCCCLXXXVIII, DCCCXCIV, DCCCXCVI, MDCCC, MDCCCIV,
>> MDCCCVI, MDCCCVIII, MDCCCXII, MDCCCXVI, MDCCCXVIII, MDCCCXX, MDCCCXXIV,
>> MDCCCXXX, MDCCCXXXII, MDCCCXXXVI, MDCCCXL, MDCCCXLVIII, MDCCCLVI, MDCCCLX,
>> MDCCCLXIV, MDCCCLXXII, MDCCCLXXVI, MDCCCLXXX, MDCCCLXXXIV, MDCCCLXXXVIII,
>> MDCCCXC, MDCCCXCVI, MMDCC, MMDCCVI, MMDCCXII, MMDCCXVIII, MMDCCXXIV, M...
>> I wrote the Boolean romNumDivAnagrammable() function to only consider the
>> divisors of *n* other than 1 and *n* itself. If I haven't made a mistake
>> somewhere, I can assert that this doesn't make a difference for "divisor
>> anagrammables" in Roman numerals. Taking this to decimal, we see that 12
>> would not thus be "divisor anagrammable," since the divisor 1 is not
>> considered. Thoughts?
>> Al
>> --
>> Alonso del Arte
>> Author at SmashWords.com
>> <https://www.smashwords.com/profile/view/AlonsoDelarte>
>> Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>> ------------------------------
>> Message: 4
>> Date: Fri, 9 Apr 2021 22:06:31 -0400
>> From: Neil Sloane <njasloane at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Anyone recognize this matrix?
>> Message-ID:
>> <CAAOnSgS89HNuj2uW5NMGx_qCPV8mu_h_Gi=wLJR-eROGQXhzVA at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> Dear Sequence Fans, I have an infinite 0,1 matrix. The first row is 01
>> repeated, the second row is 0100 repeated, and so on. Here are the first 32
>> rows.
>> I have a feeling I've seen this before, but I can't remember where.
>> I have the definition, but I would like a simple description.
>> Does anyone recognize this?
>> There are some obvious properties. In rows 8 through 15, for instance, the
>> mod 2 sums row 8 + row 15 = row 9 + row 14 = ... = row 11 + row 12 =
>> 0000000100000001.
>> And similarly for rows 2 to 3; 4 to 7; 16 to 31; etc.
>> 1: 01*
>> 2: 0100*
>> 3: 0001*
>> 4: 00010000*
>> 5: 01000101*
>> 6: 01010100*
>> 7: 00000001*
>> 8: 0000000100000000*
>> 9: 0101010001010101*
>> 10: 0100010101000100*
>> 11: 0001000000010001*
>> 12: 0001000100010000*
>> 13: 0100010001000101*
>> 14: 0101010101010100*
>> 15: 0000000000000001*
>> 16: 00000000000000010000000000000000*
>> 17: 01010101010101000101010101010101*
>> 18: 01000100010001010100010001000100*
>> 19: 00010001000100000001000100010001*
>> 20: 00010000000100010001000000010000*
>> 21: 01000101010001000100010101000101*
>> 22: 01010100010101010101010001010100*
>> 23: 00000001000000000000000100000001*
>> 24: 00000001000000010000000100000000*
>> 25: 01010100010101000101010001010101*
>> 26: 01000101010001010100010101000100*
>> 27: 00010000000100000001000000010001*
>> 28: 00010001000100010001000100010000*
>> 29: 01000100010001000100010001000101*
>> 30: 01010101010101010101010101010100*
>> 31: 00000000000000000000000000000001*
>> [These are actually the odd-numbered rows 1,3,5,7,... of the matrix. The
>> even-numbered rows have a simple formula. Row 2k is 0^(2^m) 1^(2^m)
>> repeated, where m is the number of times 2 divides 2k.
>> Row 24 for example (where m=3) is 0000000011111111 repeated. I'm hoping for
>> something similar for the odd-numbered rows.]
>> ------------------------------
>> Message: 5
>> Date: Sat, 17 Apr 2021 17:55:15 +0300
>> From: Olivier Gerard <olivier.gerard at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Recent perturbations on the Seqfan Mailing List
>> Message-ID:
>> <CAAcpPSYWHXp6ZMaMVd4-K0-_EVACNRdXgPtyAhq79Z5ywjRhrA at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> Dear subscribers of the Seqfan Mailing List,
>> This is a test message as our server has suffered from technical glitches
>> lately
>> and they may reappear and take some time to calm down.
>> Please do not respond to this message on the list, but you can send a
>> private message
>> to me if you want to.
>> With all my apologies for any inconvenience it may have caused.
>> With my best regards,
>> Olivier GERARD
>> Seqfan Mailing List Administrator
>> olivier.gerard at gmail.com
>> ------------------------------
>> Message: 6
>> Date: Thu, 15 Apr 2021 16:56:06 -0400
>> From: Alonso Del Arte <alonso.delarte at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Conjecture: a(n) = n only if n = 1 or 9
>> Message-ID:
>> <CAGyGvfUowAjOVEE8q1Ac6AhH5BNJv0wUREBKBwxV=b_ypkBBoQ at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> Given a(n) = a(n) = \prod_{i = 1}^{n - 1} gcd(i, n) (A051190) it is obvious
>> that a(n) = 1 only if n is 1 or a prime.
>> For many OEIS entries Charles has given a heuristic for growth, though not
>> for this one. It seems to be a decent fraction of n! for composite n.
>> Of course this doesn't rule out that there might be some larger n such that
>> a(n) = n.
>> Thoughts, anyone?
>> Al
>> --
>> Alonso del Arte
>> Author at SmashWords.com
>> <https://www.smashwords.com/profile/view/AlonsoDelarte>
>> Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>> ------------------------------
>> Message: 7
>> Date: Sat, 17 Apr 2021 17:45:25 +0200
>> From: Tomasz Ordowski <tomaszordowski at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Lucasian (pseudo)primes
>> Message-ID:
>> <CAF0qcNN0Q690OHqXH=FQvidTXa+hnfEYubK6fOHTDfNE1cETOA at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> Dear readers!
>> If p is a Lucasian prime, i.e.
>> p == 3 (mod 4) with 2p+1 prime;
>> then (2^p-1)/(2p+1) == 1 (mod p),
>> hence 2^p-2p-2 == 0 (mod p(2p+1)),
>> so 2^(p-1) == p+1 (mod p(2p+1)).
>> Composites k such that 2^(k-1) == k+1 (mod k(2k+1)) are
>> 150851, 452051, 1325843, 1441091, 4974971, 5016191, 15139199, 19020191,
>> 44695211, 101276579, 119378351, 128665319, 152814531, 187155383, 203789951,
>> 223782263, 307367171, 387833531, 392534231, 470579831, 505473263,
>> 546748931, 626717471, 639969891, 885510239, 974471243, 1147357559,
>> 1227474431, 1284321611, 1304553251, 1465307351, 1474936871, 1514608559,
>> 1529648231, 1639846391, 1672125131, 2117031263, 2139155051, 2304710123,
>> 2324867399, 2939179643, 3056100623, 3271076771, 3280593611, 3529864391,
>> 3587553971, 4193496803, 4244663651, 4267277291, 4278305651, 4528686251, ...
>> [Data from Amiram Eldar]
>> Conjecture:
>> These are pseudoprimes k == 3 (mod 4) such that 2k+1 is prime.
>> If so, the name "Lucasian pseudoprimes" will be fully justified.
>> There are 101629 Fermat pseudoprimes up to 10^12.
>> Of them 276 are of the type k == 3 (mod 4) with 2k+1 prime.
>> The first 51 of them are exactly those that I have sent you earlier.
>> [Amiram Eldar]
>> The conjecture is easily proved.
>> Let q = 2k+1 be prime, where k == 3 (mod 4) is a pseudoprime.
>> We have q == 7 (mod 8), so 2 is a square mod q,
>> which gives 2^{(q-1)/2} == 1 (mod q), by Euler's criterion.
>> Thus, 2^k == 1 (mod q), which implies 2^{k-1} == (q+1)/2 (mod q),
>> so that 2^{k-1} == k+1 (mod q).
>> The conclusion that 2^{k-1} == k+1 (mod kq) follows
>> from the assumption that k is a pseudoprime
>> and from the Chinese remainder theorem.
>> [Carl Pomerance]
>> Problem:
>> Are there infinitely many numbers n such that 2^{n-1} == n+1 (mod n(2n+1))
>> ?
>> These are primes and pseudoprimes n == 3 (mod 4) with 2n+1 prime.
>> It is not known whether there are infinitely many Lucasian primes.
>> Question:
>> Are there pseudoprimes m == 3 (mod 4) such that 2m+1 is a pseudoprime?
>> There only 3 known pseudoprimes m such that 2m+1 is a pseudoprime:
>> 9890881, 23456248059221, 96076792050570581 (see A303447),
>> but all the three have m == 1 (mod 4).
>> [Amiram Eldar]
>> Best regards,
>> Thomas Ordowski
>> ____________________
>> https://oeis.org/A002515
>> https://oeis.org/A081858
>> https://oeis.org/A001567
>> https://oeis.org/A303447
>> ------------------------------
>> Message: 8
>> Date: Sat, 17 Apr 2021 14:50:57 -0400
>> From: "D. S. McNeil" <dsm054 at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: Conjecture: a(n) = n only if n = 1 or 9
>> Message-ID:
>> <CAOX9QiAL2+z-L_Fb9BPRHayH4Khncr2wfC-hmOyhKR-x8u4K5Q at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> I think it's true. Sketch I'm too lazy to formalize (and whenever I break
>> things apart by cases I feel like I'm missing something obvious):
>> Say some composite n is a pure prime power p^k.
>> For k=2, we pick up a factor of p every p in the product (except for p^2
>> itself), and so A(p^2) = p^(p-1). The only solution p=3 generates the n=9
>> solution. For p > 3, A(p^2) > n.
>> For k=3, we pick up additional factors of p^2, and if I'm counting
>> correctly A(p^3) = p^(p^2+p-2), which generates no solutions. Moreover,
>> A(p^3) > n for all p.
>> For k>=4, we have at least prod(p^i,i=1..k-1) as part of the product, and
>> so p^((k-1)*k/2) | A(n). Since (k-1)*k/2 > k for k >= 4, this suffices to
>> show A(n) > n.
>> Now assume it's not a prime power, and we have n = p_0^k_0 p_1^k_1..
>> p_m^k_m. It suffices to show A(p^k*q) > p^k*q for gcd(p, q) = 1.
>> We'll pick up factors of p^k and q immediately in the expanded product, and
>> thus p^k * q | A(p^k*q), so all we need is one other component to exceed
>> p^k * q.
>> If k>=2, p itself contributes in the terms: [p, p^k, q] giving p^(k+1) * q
>> | A(n).
>> If k = 1 and p=2, then q >= 3, n >= 6, and thus [p=2, 2*p=4, q] giving
>> p^2*q | A(n).
>> If k = 1 and q=2, then p >= 3, n >= 6, and thus [p, q=2, 2*q=4] giving
>> p*q^2 | A(n)
>> If k = 1 and p,q >= 3, then we have [p, 2*p, q] giving p^2 * q| A(n)
>> And I think that's all the cases.
>> If I didn't make a silly error in the above it'd be the first time ever,
>> but I'm pretty sure some argument along these lines works.
>> Doug #lockdownboredom
>> ------------------------------
>> Message: 9
>> Date: Sat, 17 Apr 2021 17:37:37 +0000 (UTC)
>> From: Frank Adams-watters <franktaw at netscape.net>
>> To: "seqfan at seqfan.eu" <seqfan at seqfan.eu>
>> Subject: [seqfan] Sum-Product Problem
>> Message-ID: <499880394.1809449.1618681057196 at mail.yahoo.com>
>> Content-Type: text/plain; charset=UTF-8
>> There's an article on this problem in Quanta magazine:
>> https://www.quantamagazine.org/the-sum-product-problem-shows-how-addition-and-multiplication-constrain-each-other-20190206/
>> Should this be added as a link to A027424? Or do the academic links
>> already there suffice?
>> Franklin T. Adams-Watters
>> ------------------------------
>> Message: 10
>> Date: Wed, 14 Apr 2021 21:08:03 -0400
>> From: Allan Wechsler <acwacw at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Planar distributive lattices
>> Message-ID:
>> <CADy-sGHf=VWa90-vsBdqRTOCPFCNVL1B6ECknaOb7Xgr7Tt+9w at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> I forget how I stumbled on this:
>> https://math.chapman.edu/~jipsen/mathposters/Planar%20distributive%20lattices%20up%20to%20size%2011.pdf
>> .
>> It is a chart purporting to show all of the planar distributive lattices
>> with up to 11 vertices. Like any true-hearted sequence fanatic I counted
>> the number of these guys of each order, and got the following sequence:
>> 1,1,1,2,3,5,8,14,24,42,72...
>> Imagine my surprise at finding this sequence missing from OEIS! The author
>> is apparently Dr. Peter Jipsen, at Chapman University in California.
>> Perhaps someone here can figure out (a) what a planar distributive lattice
>> is, (b) whether Dr. Jipsen enumerated them correctly, (c) whether I counted
>> them off Jipsen's poster correctly, and (d) whether to add the sequence.
>> Thank you!
>> ------------------------------
>> Message: 11
>> Date: Sat, 17 Apr 2021 21:23:05 -0400
>> From: Neil Sloane <njasloane at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Cc: "seqfan at seqfan.eu" <seqfan at seqfan.eu>, Frank Adams-watters
>> <franktaw at netscape.net>
>> Subject: [seqfan] Re: Sum-Product Problem
>> Message-ID:
>> <CAAOnSgSHELbp3auLvoS-DPK6bYbOnqTPWX8xj=1CJneBXa03eg at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> certainly add it - thanks!
>> Best regards
>> Neil
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>> On Sat, Apr 17, 2021 at 9:21 PM Frank Adams-watters via SeqFan <
>> seqfan at list.seqfan.eu> wrote:
>>> There's an article on this problem in Quanta magazine:
>>> https://www.quantamagazine.org/the-sum-product-problem-shows-how-addition-and-multiplication-constrain-each-other-20190206/
>>> Should this be added as a link to A027424? Or do the academic links
>>> already there suffice?
>>> Franklin T. Adams-Watters
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> ------------------------------
>> Message: 12
>> Date: Sat, 17 Apr 2021 21:23:05 -0400
>> From: Neil Sloane <njasloane at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Cc: "seqfan at seqfan.eu" <seqfan at seqfan.eu>, Frank Adams-watters
>> <franktaw at netscape.net>
>> Subject: [seqfan] Re: Sum-Product Problem
>> Message-ID:
>> <CAAOnSgSHELbp3auLvoS-DPK6bYbOnqTPWX8xj=1CJneBXa03eg at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> certainly add it - thanks!
>> Best regards
>> Neil
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>> On Sat, Apr 17, 2021 at 9:21 PM Frank Adams-watters via SeqFan <
>> seqfan at list.seqfan.eu> wrote:
>>> There's an article on this problem in Quanta magazine:
>>> https://www.quantamagazine.org/the-sum-product-problem-shows-how-addition-and-multiplication-constrain-each-other-20190206/
>>> Should this be added as a link to A027424? Or do the academic links
>>> already there suffice?
>>> Franklin T. Adams-Watters
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> ------------------------------
>> Message: 13
>> Date: Sun, 18 Apr 2021 01:17:36 -0400
>> From: Neil Sloane <njasloane at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: Planar distributive lattices
>> Message-ID:
>> <CAAOnSgQDtfv9ZpBauaYJqPBveVKc=98pZ00rY_uKZ3f=g=_MwA at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> Allan, I confirm your numbers. I'll run it through Superseeker, which will
>> tell us if it is a simple cousin of an existing entry, and if it doesn;t
>> find anything I will add it - it will be A343161.
>> Thanks for catching this fish!
>> Best regards
>> Neil
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>> On Sat, Apr 17, 2021 at 9:22 PM Allan Wechsler <acwacw at gmail.com> wrote:
>>> I forget how I stumbled on this:
>>> https://math.chapman.edu/~jipsen/mathposters/Planar%20distributive%20lattices%20up%20to%20size%2011.pdf
>>> .
>>> It is a chart purporting to show all of the planar distributive lattices
>>> with up to 11 vertices. Like any true-hearted sequence fanatic I counted
>>> the number of these guys of each order, and got the following sequence:
>>> 1,1,1,2,3,5,8,14,24,42,72...
>>> Imagine my surprise at finding this sequence missing from OEIS! The author
>>> is apparently Dr. Peter Jipsen, at Chapman University in California.
>>> Perhaps someone here can figure out (a) what a planar distributive lattice
>>> is, (b) whether Dr. Jipsen enumerated them correctly, (c) whether I counted
>>> them off Jipsen's poster correctly, and (d) whether to add the sequence.
>>> Thank you!
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> ------------------------------
>> Message: 14
>> Date: Sun, 18 Apr 2021 01:38:57 -0400
>> From: Neil Sloane <njasloane at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: Planar distributive lattices
>> Message-ID:
>> <CAAOnSgR_QP5B7hMVB7KZ+=tdP4eSPqFoYnhVd7+6PfiDeXvT=Q at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> PS I see that the author of that poster updated it in 2014, extending it to
>> 15 vertices:
>> https://math.chapman.edu/~jipsen/tikzsvg/planar-distributive-lattices15.html
>> I am going to write to him
>> Best regards
>> Neil
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>> On Sun, Apr 18, 2021 at 1:17 AM Neil Sloane <njasloane at gmail.com> wrote:
>>> Allan, I confirm your numbers. I'll run it through Superseeker, which
>>> will tell us if it is a simple cousin of an existing entry, and if it
>>> doesn;t find anything I will add it - it will be A343161.
>>> Thanks for catching this fish!
>>> Best regards
>>> Neil
>>> Neil J. A. Sloane, President, OEIS Foundation.
>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>>> Phone: 732 828 6098; home page: http://NeilSloane.com
>>> Email: njasloane at gmail.com
>>> On Sat, Apr 17, 2021 at 9:22 PM Allan Wechsler <acwacw at gmail.com> wrote:
>>>> I forget how I stumbled on this:
>>>> https://math.chapman.edu/~jipsen/mathposters/Planar%20distributive%20lattices%20up%20to%20size%2011.pdf
>>>> .
>>>> It is a chart purporting to show all of the planar distributive lattices
>>>> with up to 11 vertices. Like any true-hearted sequence fanatic I counted
>>>> the number of these guys of each order, and got the following sequence:
>>>> 1,1,1,2,3,5,8,14,24,42,72...
>>>> Imagine my surprise at finding this sequence missing from OEIS! The author
>>>> is apparently Dr. Peter Jipsen, at Chapman University in California.
>>>> Perhaps someone here can figure out (a) what a planar distributive lattice
>>>> is, (b) whether Dr. Jipsen enumerated them correctly, (c) whether I
>>>> counted
>>>> them off Jipsen's poster correctly, and (d) whether to add the sequence.
>>>> Thank you!
>>>> --
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>> ------------------------------
>> Message: 15
>> Date: Sun, 18 Apr 2021 11:10:42 +0200
>> From: Hugo Pfoertner <yae9911 at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: Sum-Product Problem
>> Message-ID:
>> <CAEoHttX4cE=oND4TxkevV-6knbq1=QVntzDBiBQfd2afNELtnQ at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> A263996 <https://oeis.org/A263996> is even more closely related. I also
>> added the link there.
>> On Sun, Apr 18, 2021 at 3:23 AM Neil Sloane <njasloane at gmail.com> wrote:
>>> certainly add it - thanks!
>>> Best regards
>>> Neil
>>> Neil J. A. Sloane, President, OEIS Foundation.
>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>>> Phone: 732 828 6098; home page: http://NeilSloane.com
>>> Email: njasloane at gmail.com
>>> On Sat, Apr 17, 2021 at 9:21 PM Frank Adams-watters via SeqFan <
>>> seqfan at list.seqfan.eu> wrote:
>>> > There's an article on this problem in Quanta magazine:
>>> >
>>> >
>>> >
>>> https://www.quantamagazine.org/the-sum-product-problem-shows-how-addition-and-multiplication-constrain-each-other-20190206/
>>> >
>>> > Should this be added as a link to A027424? Or do the academic links
>>> > already there suffice?
>>> >
>>> > Franklin T. Adams-Watters
>>> >
>>> > --
>>> > Seqfan Mailing list - http://list.seqfan.eu/
>>> >
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> ------------------------------
>> Message: 16
>> Date: Sun, 18 Apr 2021 14:05:21 -0400
>> From: Alonso Del Arte <alonso.delarte at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: Conjecture: a(n) = n only if n = 1 or 9
>> Message-ID:
>> <CAGyGvfXhq4Ae4xRXfF=mK5UGjs4aMTV7zDqOek=yrnQ5Jk0xVg at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> Thanks, D. S., I think it checks out. I appreciate your taking the time to
>> sketch this out. In all honesty, the most I did was scan the B-file.
>> Al
>> On Sat, Apr 17, 2021 at 2:51 PM D. S. McNeil <dsm054 at gmail.com> wrote:
>>> I think it's true. Sketch I'm too lazy to formalize (and whenever I break
>>> things apart by cases I feel like I'm missing something obvious):
>>> Say some composite n is a pure prime power p^k.
>>> For k=2, we pick up a factor of p every p in the product (except for p^2
>>> itself), and so A(p^2) = p^(p-1). The only solution p=3 generates the n=9
>>> solution. For p > 3, A(p^2) > n.
>>> For k=3, we pick up additional factors of p^2, and if I'm counting
>>> correctly A(p^3) = p^(p^2+p-2), which generates no solutions. Moreover,
>>> A(p^3) > n for all p.
>>> For k>=4, we have at least prod(p^i,i=1..k-1) as part of the product, and
>>> so p^((k-1)*k/2) | A(n). Since (k-1)*k/2 > k for k >= 4, this suffices to
>>> show A(n) > n.
>>> Now assume it's not a prime power, and we have n = p_0^k_0 p_1^k_1..
>>> p_m^k_m. It suffices to show A(p^k*q) > p^k*q for gcd(p, q) = 1.
>>> We'll pick up factors of p^k and q immediately in the expanded product, and
>>> thus p^k * q | A(p^k*q), so all we need is one other component to exceed
>>> p^k * q.
>>> If k>=2, p itself contributes in the terms: [p, p^k, q] giving p^(k+1) * q
>>> | A(n).
>>> If k = 1 and p=2, then q >= 3, n >= 6, and thus [p=2, 2*p=4, q] giving
>>> p^2*q | A(n).
>>> If k = 1 and q=2, then p >= 3, n >= 6, and thus [p, q=2, 2*q=4] giving
>>> p*q^2 | A(n)
>>> If k = 1 and p,q >= 3, then we have [p, 2*p, q] giving p^2 * q| A(n)
>>> And I think that's all the cases.
>>> If I didn't make a silly error in the above it'd be the first time ever,
>>> but I'm pretty sure some argument along these lines works.
>>> Doug #lockdownboredom
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> --
>> Alonso del Arte
>> Author at SmashWords.com
>> <https://www.smashwords.com/profile/view/AlonsoDelarte>
>> Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>> ------------------------------
>> Message: 17
>> Date: Sun, 18 Apr 2021 17:36:45 -0400
>> From: Allan Wechsler <acwacw at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: Planar distributive lattices
>> Message-ID:
>> <CADy-sGG_avdNiEmVMj2=8NM34CiVRO_O=SRWWaE9suFNJwzo=A at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> I think writing to him is a great idea; get him to author that sequence,
>> which he only deserves, after all.
>> On Sun, Apr 18, 2021 at 1:39 AM Neil Sloane <njasloane at gmail.com> wrote:
>>> PS I see that the author of that poster updated it in 2014, extending it to
>>> 15 vertices:
>>> https://math.chapman.edu/~jipsen/tikzsvg/planar-distributive-lattices15.html
>>> I am going to write to him
>>> Best regards
>>> Neil
>>> Neil J. A. Sloane, President, OEIS Foundation.
>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>>> Phone: 732 828 6098; home page: http://NeilSloane.com
>>> Email: njasloane at gmail.com
>>> On Sun, Apr 18, 2021 at 1:17 AM Neil Sloane <njasloane at gmail.com> wrote:
>>> > Allan, I confirm your numbers. I'll run it through Superseeker, which
>>> > will tell us if it is a simple cousin of an existing entry, and if it
>>> > doesn;t find anything I will add it - it will be A343161.
>>> >
>>> > Thanks for catching this fish!
>>> >
>>> > Best regards
>>> > Neil
>>> >
>>> > Neil J. A. Sloane, President, OEIS Foundation.
>>> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>> > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>>> > Phone: 732 828 6098; home page: http://NeilSloane.com
>>> > Email: njasloane at gmail.com
>>> >
>>> >
>>> >
>>> > On Sat, Apr 17, 2021 at 9:22 PM Allan Wechsler <acwacw at gmail.com> wrote:
>>> >
>>> >> I forget how I stumbled on this:
>>> >>
>>> >>
>>> https://math.chapman.edu/~jipsen/mathposters/Planar%20distributive%20lattices%20up%20to%20size%2011.pdf
>>> >> .
>>> >>
>>> >> It is a chart purporting to show all of the planar distributive lattices
>>> >> with up to 11 vertices. Like any true-hearted sequence fanatic I counted
>>> >> the number of these guys of each order, and got the following sequence:
>>> >>
>>> >> 1,1,1,2,3,5,8,14,24,42,72...
>>> >>
>>> >> Imagine my surprise at finding this sequence missing from OEIS! The
>>> author
>>> >> is apparently Dr. Peter Jipsen, at Chapman University in California.
>>> >>
>>> >> Perhaps someone here can figure out (a) what a planar distributive
>>> lattice
>>> >> is, (b) whether Dr. Jipsen enumerated them correctly, (c) whether I
>>> >> counted
>>> >> them off Jipsen's poster correctly, and (d) whether to add the sequence.
>>> >>
>>> >> Thank you!
>>> >>
>>> >> --
>>> >> Seqfan Mailing list - http://list.seqfan.eu/
>>> >>
>>> >
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> ------------------------------
>> Message: 18
>> Date: Sun, 18 Apr 2021 22:10:06 -0400
>> From: Neil Sloane <njasloane at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: Planar distributive lattices
>> Message-ID:
>> <CAAOnSgRb+u4ToEYxTM1+F2ns-7WHoQFr6vmsf=GnF8f0MhguLg at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> Allan Wechsler said, in connection with the sequence I created (A343161)
>> based on Peter Jipsen's enumeration: "I think writing to him is a great
>> idea; get him to author that sequence, which he only deserves, after all."
>> Well, he has been a registered user - and contributor - to the OEIS since
>> March 2013. He never submitted it, and Allan wasn't sure it should be in
>> the OEIS, so I created it. Today I extended it to 15 terms using the data
>> in Peter's 2014 pdf file.
>> Incidentally Peter thanked me for creating the entry and said in reply that
>> he will try to find his old program and compute more terms.
>> Best regards
>> Neil
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>> On Sun, Apr 18, 2021 at 7:52 PM Allan Wechsler <acwacw at gmail.com> wrote:
>>> I think writing to him is a great idea; get him to author that sequence,
>>> which he only deserves, after all.
>>> On Sun, Apr 18, 2021 at 1:39 AM Neil Sloane <njasloane at gmail.com> wrote:
>>> > PS I see that the author of that poster updated it in 2014, extending it
>>> to
>>> > 15 vertices:
>>> >
>>> >
>>> >
>>> https://math.chapman.edu/~jipsen/tikzsvg/planar-distributive-lattices15.html
>>> >
>>> > I am going to write to him
>>> >
>>> >
>>> > Best regards
>>> > Neil
>>> >
>>> > Neil J. A. Sloane, President, OEIS Foundation.
>>> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>> > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>>> > Phone: 732 828 6098; home page: http://NeilSloane.com
>>> > Email: njasloane at gmail.com
>>> >
>>> >
>>> >
>>> > On Sun, Apr 18, 2021 at 1:17 AM Neil Sloane <njasloane at gmail.com> wrote:
>>> >
>>> > > Allan, I confirm your numbers. I'll run it through Superseeker, which
>>> > > will tell us if it is a simple cousin of an existing entry, and if it
>>> > > doesn;t find anything I will add it - it will be A343161.
>>> > >
>>> > > Thanks for catching this fish!
>>> > >
>>> > > Best regards
>>> > > Neil
>>> > >
>>> > > Neil J. A. Sloane, President, OEIS Foundation.
>>> > > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>> > > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway,
>>> NJ.
>>> > > Phone: 732 828 6098; home page: http://NeilSloane.com
>>> > > Email: njasloane at gmail.com
>>> > >
>>> > >
>>> > >
>>> > > On Sat, Apr 17, 2021 at 9:22 PM Allan Wechsler <acwacw at gmail.com>
>>> wrote:
>>> > >
>>> > >> I forget how I stumbled on this:
>>> > >>
>>> > >>
>>> >
>>> https://math.chapman.edu/~jipsen/mathposters/Planar%20distributive%20lattices%20up%20to%20size%2011.pdf
>>> > >> .
>>> > >>
>>> > >> It is a chart purporting to show all of the planar distributive
>>> lattices
>>> > >> with up to 11 vertices. Like any true-hearted sequence fanatic I
>>> counted
>>> > >> the number of these guys of each order, and got the following
>>> sequence:
>>> > >>
>>> > >> 1,1,1,2,3,5,8,14,24,42,72...
>>> > >>
>>> > >> Imagine my surprise at finding this sequence missing from OEIS! The
>>> > author
>>> > >> is apparently Dr. Peter Jipsen, at Chapman University in California.
>>> > >>
>>> > >> Perhaps someone here can figure out (a) what a planar distributive
>>> > lattice
>>> > >> is, (b) whether Dr. Jipsen enumerated them correctly, (c) whether I
>>> > >> counted
>>> > >> them off Jipsen's poster correctly, and (d) whether to add the
>>> sequence.
>>> > >>
>>> > >> Thank you!
>>> > >>
>>> > >> --
>>> > >> Seqfan Mailing list - http://list.seqfan.eu/
>>> > >>
>>> > >
>>> >
>>> > --
>>> > Seqfan Mailing list - http://list.seqfan.eu/
>>> >
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> ------------------------------
>> Message: 19
>> Date: Sun, 18 Apr 2021 21:54:25 -0700
>> From: peter lawrence <peterl95124 at comcast.net>
>> To: seqfan at list.seqfan.eu
>> Cc: Olivier Gerard <olivier.gerard at gmail.com>, Susanna Cuyler
>> <Susanna.Cuyler at gmail.com>, Neil Sloane <njasloane at gmail.com>
>> Subject: [seqfan] help naming/describing sequences for bounds of
>> Goldbach's Comet, was: what happened to my A342302, its been replaced
>> ???
>> Message-ID: <5B52ED1E-B02E-42B9-A71C-7E5AA36BCEE5 at comcast.net>
>> Content-Type: text/plain; charset=utf-8
>> All,
>> I agree that the encyclopedia itself maybe isn?t the right place
>> to work out the details,
>> so lets do that here in seqfan instead.
>> Goldbach's Comet (e.g.
>> https://en.wikipedia.org/wiki/Goldbach%27s_comet
>> <https://en.wikipedia.org/wiki/Goldbach's_comet>, and lots of google
>> images, etc)
>> has obvious upper and lower bounds, at least visually, so the natural
>> question is what are those bounds.
>> I?m using A002372 for Goldbach counts (order dependent sums) because
>> they can be computed as a convolution,
>> and because there?s something obvious about when the value is even
>> verse odd that is much less easy to state
>> when using A002375 (unordered sums).
>> note that A002372 isn't indexed by N, rather it's indexed by N/2
>> because odd numbers aren?t generally
>> the sum of two primes (in general only the upper half of a twin prime
>> pair is), and the factor of two
>> makes it awkward to relate that sequence to ones discussed here where
>> I talk about what N achieves a
>> bound rather than what N/2.
>> it would be nice to have approximation formulas,
>> and it would also be nice to have sequences that could be found in OEIS,
>> I?d like to work on both, but for this email I?ll stick with sequences.
>> for the upper bound sequence one might tabulate where a high point is
>> first achieved, as in
>> 1 is first achieved at N = 6, 6 = {3+3} = 1 way
>> 2 is first achieved at N = 8, 8 = {3+5, 5+3} = 2 ways
>> 3 is first achieved at N = 10, 10 = {3+7, 5+5, 7+3} = 3 ways
>> 4 is first achieved at N = 16, 16 = {3+13, 5+11, 11+5, 13+3} = 4 ways
>> 5 is first achieved at N = 22, 22 = {3+19, 5+17, 11+11, 17+5, 19+3} = 5 ways
>> etc...
>> so 6,8,10,16,22,... maybe should be in OEIS ?
>> well not so fast, continuing we find that the first and last N for
>> each possible count 1..20 are
>> count first last
>> 1 6 6
>> 2 8 12
>> 3 10 38
>> 4 16 68
>> 5 22 62
>> 6 24 128
>> 7 34 122
>> 8 36 152
>> 9 74 158 --- 9 isn?t achieved until N=74, but a couple
>> higher counts (10,120 are achieved earlier
>> 10 48 188
>> 11 106 166 --- ditto
>> 12 60 332
>> 13 178 398 --- and so it goes for all odd counts since an
>> odd count only
>> 14 78 272
>> 15 142 362 --- occurs for N = 2 x Prime which are more rare
>> than N = 2 x Composite
>> 16 84 368
>> 17 202 458
>> 18 90 488
>> 19 358 542
>> 20 114 632
>> so the strict upper bounds (points on the (top side of the) convex
>> hull of Goldbach?s Comet)
>> don?t include all possible counts, example ?9? above.
>> also note that for just even counts the first occurrence isn?t always
>> steadily increasing
>> 30 234 908
>> 32 246 1112
>> 34 288 968
>> 36 240 1412 ? the count 36 occurs earlier than 32, and 34
>> 38 210 1178 ? the count 38 occurs earlier than 30, 32, 34, and 36
>> 40 324 1448
>> so not all even counts (which in general occur earlier than odd
>> counts) are on the (top
>> side of the) convex hull either.
>> similarly the ?last occurrence? sequence contains points not on the
>> (bottom side of the) convex
>> hull of Goldbach?s Comet.
>> [One might ask, what is the best way to put points on the convex hull
>> of Goldbach?s Comet into OEIS,
>> if you have a suggestion feel free to comment, those are what I
>> consider the true upper and lower
>> bounds, but those are pairs of integers. Yes they are integers, yes
>> they can be sequenced, but they
>> are pairs. Would you create two (related) sequences ? I?m not
>> proposing anything on that topic,
>> instead...]
>> but even though the above sequences aren?t the lower and upper bounds
>> I was looking for,
>> and one might call them longitudinal studies of Goldbach?s Comet instead,
>> they still seem to be of sufficient mathematical interest to be in OEIS, because
>> There are some obvious conjectures
>> 1. every count is the Goldbach count for some N
>> equivalently every number occurs in A002372 (there is a first occurrence)
>> 2. every count is the Goldbach count for only finitely many N,
>> equivalently there is a last occurrence in A002372 for each number
>> so I hope folks on this list can help with viable descriptions for
>> these sequences
>> such that they will be accepted into the OEIS
>> what would you name these sequences
>> (I consider them well defined, IMHO, but) how would you describe these sequences
>> if you don?t think they are well defined please state why
>> Peter A Lawrence
>>> On Apr 18, 2021, at 6:51 PM, Neil Sloane <njasloane at gmail.com> wrote:
>>> Peter Lawrence,
>>> Following the advice of some senior editors - but based primarily on my own views - your sequence was rejected by me on April 2 2021 as "interesting but not ready for the OEIS". You would have a received a copy of this decision since you were the author.
>>> Furthermore, this was also recorded in the webpage on the OEIS Wiki called Deleted Sequences.
>>> The reasons are documented in the Pink Box discussions of the sequence, which you can see by going to the "history" tab of A342302.
>>> Best regards
>>> Neil
>>> Neil J. A. Sloane, President, OEIS Foundation.
>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>>> Phone: 732 828 6098; home page: http://NeilSloane.com <http://neilsloane.com/>
>>> Email: njasloane at gmail.com <mailto:njasloane at gmail.com>
>> ------------------------------
>> Message: 20
>> Date: Wed, 21 Apr 2021 14:47:35 +0200
>> From: "Richard J. Mathar" <mathar at mpia-hd.mpg.de>
>> To: seqfan at list.seqfan.eu
>> Subject: [seqfan] Re: Anyone recognize this matrix?
>> Message-ID: <20210421124735.GA20127 at mathar.mpia-hd.mpg.de>
>> Content-Type: text/plain; charset=us-ascii
>> A formal description of this infinite array of 0's and 1's is:
>> The "full" array including a leading row of all-0 starts as follows:
>> 0 00000000000000000
>> 1 01010101010101010
>> 2 01000100010001000
>> 3 00010001000100010
>> 4 00010000000100000
>> 5 01000101010001010
>> 6 01010100010101000
>> 7 00000001000000010
>> 8 00000001000000000
>> 9 01010100010101010
>> 10 01000101010001000
>> 11 00010000000100010
>> 12 00010001000100000
>> 13 01000100010001010
>> 14 01010101010101000
>> 15 00000000000000010
>> 16 00000000000000010
>> 17 01010101010101000
>> 18 01000100010001010
>> 19 00010001000100000
>> 20 00010000000100010
>> 21 01000101010001000
>> 22 01010100010101010
>> 23 00000001000000000
>> 24 00000001000000010
>> 25 01010100010101000
>> 26 01000101010001010
>> 27 00010000000100000
>> 28 00010001000100010
>> 29 01000100010001000
>> 30 01010101010101010
>> 31 00000000000000000
>> Because each second column contains only zeros, we delete each second column
>> and get the "reduced" array
>> 0 00000000000000000
>> 1 11111111111111111
>> 2 10101010101010101
>> 3 01010101010101010
>> 4 01000100010001000
>> 5 10111011101110111
>> 6 11101110111011101
>> 7 00010001000100010
>> 8 00010000000100000
>> 9 11101111111011111
>> 10 10111010101110101
>> 11 01000101010001010
>> 12 01010100010101000
>> 13 10101011101010111
>> 14 11111110111111101
>> 15 00000001000000010
>> 16 00000001000000000
>> 17 11111110111111111
>> 18 10101011101010101
>> 19 01010100010101010
>> 20 01000101010001000
>> 21 10111010101110111
>> 22 11101111111011101
>> 23 00010000000100010
>> 24 00010001000100000
>> 25 11101110111011111
>> 26 10111011101110101
>> 27 01000100010001010
>> 28 01010101010101000
>> 29 10101010101010111
>> 30 11111111111111101
>> 31 00000000000000010
>> Each odd-numbered row is the binary complement of its preceding row, so
>> define a "depleted reduced" array just containing rows 0,2,4,6,8,...:
>> 0 00000000000000000
>> 2 10101010101010101
>> 4 01000100010001000
>> 6 11101110111011101
>> 8 00010000000100000
>> 10 10111010101110101
>> 12 01010100010101000
>> 14 11111110111111101
>> 16 00000001000000000
>> 18 10101011101010101
>> 20 01000101010001000
>> 22 11101111111011101
>> 24 00010001000100000
>> 26 10111011101110101
>> 28 01010101010101000
>> 30 11111111111111101
>> The definition of this seems to be given by reading the 1st, 2nd, 3rd
>> ... column downwards, which gives periodic patterns of zeros and ones:
>> 0,1 (col 1)
>> 0,0,1,1 (col 2)
>> 0,1 (col 3)
>> 0,0,0,0,1,1,1,1 (col 4)
>> 0,1 (col 5)
>> 0,0,1,1 (col 6)
>> 0,1 (col 7)
>> 0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1 (col 8)
>> 0,1 (col 9)
>> 0,0,1,1 (col 10)
>> 0,1 (col 11)
>> 0,0,0,0,1,1,1,1 (col 12)
>> 0,1 (col 13)
>> 0,0,1,1 (col 14)
>> 0,1 (col 15)
>> 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, (col 16)
>> where the number of zeros (and number of ones) in the periods
>> of column k is given by A006519(k).
>> ------------------------------
>> Message: 21
>> Date: Wed, 21 Apr 2021 09:07:21 -0400
>> From: Neil Sloane <njasloane at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: Anyone recognize this matrix?
>> Message-ID:
>> <CAAOnSgR1XuMEHXrhCRSXK7ujKXfTMMjzyZUKQL1_gm3SwDo__Q at mail.gmail.com>
>> Content-Type: text/plain; charset="UTF-8"
>> Richard, You are right, and indeed one can say much more. In fact this is
>> part of a bigger investigation and there is a paper in progress that will
>> reveal everything. I was hoping to have it finished a week ago, but keeping
>> the OEIS running takes a great deal of time. Once the paper is in readable
>> form I will post a link to it here.
>> Best regards
>> Neil
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>> On Wed, Apr 21, 2021 at 8:47 AM Richard J. Mathar <mathar at mpia-hd.mpg.de>
>> wrote:
>>> A formal description of this infinite array of 0's and 1's is:
>>> The "full" array including a leading row of all-0 starts as follows:
>>> 0 00000000000000000
>>> 1 01010101010101010
>>> 2 01000100010001000
>>> 3 00010001000100010
>>> 4 00010000000100000
>>> 5 01000101010001010
>>> 6 01010100010101000
>>> 7 00000001000000010
>>> 8 00000001000000000
>>> 9 01010100010101010
>>> 10 01000101010001000
>>> 11 00010000000100010
>>> 12 00010001000100000
>>> 13 01000100010001010
>>> 14 01010101010101000
>>> 15 00000000000000010
>>> 16 00000000000000010
>>> 17 01010101010101000
>>> 18 01000100010001010
>>> 19 00010001000100000
>>> 20 00010000000100010
>>> 21 01000101010001000
>>> 22 01010100010101010
>>> 23 00000001000000000
>>> 24 00000001000000010
>>> 25 01010100010101000
>>> 26 01000101010001010
>>> 27 00010000000100000
>>> 28 00010001000100010
>>> 29 01000100010001000
>>> 30 01010101010101010
>>> 31 00000000000000000
>>> Because each second column contains only zeros, we delete each second
>>> column
>>> and get the "reduced" array
>>> 0 00000000000000000
>>> 1 11111111111111111
>>> 2 10101010101010101
>>> 3 01010101010101010
>>> 4 01000100010001000
>>> 5 10111011101110111
>>> 6 11101110111011101
>>> 7 00010001000100010
>>> 8 00010000000100000
>>> 9 11101111111011111
>>> 10 10111010101110101
>>> 11 01000101010001010
>>> 12 01010100010101000
>>> 13 10101011101010111
>>> 14 11111110111111101
>>> 15 00000001000000010
>>> 16 00000001000000000
>>> 17 11111110111111111
>>> 18 10101011101010101
>>> 19 01010100010101010
>>> 20 01000101010001000
>>> 21 10111010101110111
>>> 22 11101111111011101
>>> 23 00010000000100010
>>> 24 00010001000100000
>>> 25 11101110111011111
>>> 26 10111011101110101
>>> 27 01000100010001010
>>> 28 01010101010101000
>>> 29 10101010101010111
>>> 30 11111111111111101
>>> 31 00000000000000010
>>> Each odd-numbered row is the binary complement of its preceding row, so
>>> define a "depleted reduced" array just containing rows 0,2,4,6,8,...:
>>> 0 00000000000000000
>>> 2 10101010101010101
>>> 4 01000100010001000
>>> 6 11101110111011101
>>> 8 00010000000100000
>>> 10 10111010101110101
>>> 12 01010100010101000
>>> 14 11111110111111101
>>> 16 00000001000000000
>>> 18 10101011101010101
>>> 20 01000101010001000
>>> 22 11101111111011101
>>> 24 00010001000100000
>>> 26 10111011101110101
>>> 28 01010101010101000
>>> 30 11111111111111101
>>> The definition of this seems to be given by reading the 1st, 2nd, 3rd
>>> ... column downwards, which gives periodic patterns of zeros and ones:
>>> 0,1 (col 1)
>>> 0,0,1,1 (col 2)
>>> 0,1 (col 3)
>>> 0,0,0,0,1,1,1,1 (col 4)
>>> 0,1 (col 5)
>>> 0,0,1,1 (col 6)
>>> 0,1 (col 7)
>>> 0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1 (col 8)
>>> 0,1 (col 9)
>>> 0,0,1,1 (col 10)
>>> 0,1 (col 11)
>>> 0,0,0,0,1,1,1,1 (col 12)
>>> 0,1 (col 13)
>>> 0,0,1,1 (col 14)
>>> 0,1 (col 15)
>>> 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, (col 16)
>>> where the number of zeros (and number of ones) in the periods
>>> of column k is given by A006519(k).
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> ------------------------------
>> Subject: Digest Footer
>> _______________________________________________
>> SeqFan mailing list
>> SeqFan at list.seqfan.eu
>> http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan
>> ------------------------------
>> End of SeqFan Digest, Vol 151, Issue 1
>> **************************************
>
> --
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
More information about the SeqFan
mailing list