[seqfan] Re: A344826

Wed Jun 16 11:07:05 CEST 2021

Here are terms 91-133 up to 2^31-1.
The largest prime used is now 199.

/Lars B

91	208569600	2^8 3^3 5^2 17 71  
92	240997680	2^4 3^5 5 7^2 11 23 	
93	265075200	2^9 3^3 5^2 13 59 
94	269438400	2^6 3^7 5^2 7 11 	
95	298306800	2^4 3^7 5^2 11 31 
96	305363520	2^6 3^6 5 7 11 17 
97	315043344	2^4 3^4 7^2 11^2 41 	
98	321330240	2^6 3^4 5 7^2 11 23 	
99	326172672	2^16 3^2 7 79 
100	335619072	2^10 3^3 61 199 <--
101	351168048	2^4 3^6 7 11 17 23 
102	356756400	2^4 3^4 5^2 7 11^2 13 	
103	438177600	2^6 3^5 5^2 7^2 23 
104	541172016	2^4 3^4 7 11^2 17 29 	
105	585280080	2^4 3^5 5 7 11 17 23 	
106	601171200	2^8 3^3 5^2 7^2 71 
107	618508800	2^9 3^2 5^2 7 13 59 	
108	689195520	2^9 3^3 5 13^2 59 	
109	708879600	2^4 3^6 5^2 11 13 17 	
110	768398400	2^6 3^4 5^2 7^2 11^2 	
111	839749680	2^4 3^6 5 7 11^2 17
112	860569920	2^6 3^6 5 7 17 31 
113	876355200	2^7 3^5 5^2 7^2 23
114	963990720	2^6 3^5 5 7^2 11 23
115	978518016	2^16 3^3 7 79
116	1015383600	2^4 3^4 5^2 7 11^2 37
117	1069153344	2^6 3^5 7^2 23 61
118	1204988400	2^4 3^5 5^2 7^2 11 23 
119	1205452800	2^14 3^3 5^2 109 
120	1273405536	2^5 3^6 13^2 17 19
121	1494158400	2^6 3^7 5^2 7 61
122	1502092800	2^9 3^2 5^2 13 17 59
123	1575216720	2^4 3^4 5 7^2 11^2 41
124	1630863360	2^16 3^2 5 7 79
125	1657145952	2^5 3^4 7 11 19^2 23
126	1678095360	2^10 3^3 5 61 199 <--
127	1755840240	2^4 3^6 5 7 11 17 23
128	1804420800	2^6 3^6 5^2 7 13 17
129	1855526400	2^9 3^3 5^2 7 13 59
130	1975276800	2^8 3^3 5^2 7 23 71
131	1976180976	2^4 3^5 7^2 11 23 41
132	2017580400	2^4 3^6 5^2 11 17 37
133	2113095600	2^4 3^4 5^2 7^2 11^3

-----Ursprungligt meddelande-----
Från: SeqFan <seqfan-bounces at list.seqfan.eu> För M. F. Hasler
Skickat: den 16 juni 2021 01:45
Till: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Ämne: [seqfan] Re: A344826

On Tue, Jun 15, 2021 at 10:08 AM David Corneth <davidacorneth at gmail.com>
wrote:

> (...)

Maybe with some refining this method could speed the search up a little to
> at least get upperbounds of other terms or maybe even actual proven 
> new terms.
>

It's not difficult to construct new terms through the method alluded to in the COMMENTS, which I've tried made a little more precise/explicit through a new comment (pending edit):
If k = a(n) with q = A343886(k) = k/A097621(k), and m=A961(q) is relatively prime to k, then A097621(k*m) = A097621(k) * A097621(m) = A097621(k) * q so  k*m / A097621(k*m) = q * m / q = m  is an integer and k*m is in the sequence.

One might call "primitive" those terms that are not derived in this way from smaller terms.
All the terms listed so far are primitive, but 5 of the last 9 terms, a({35, 36, 38, 42, 43} = {694008, 1097712, 1778400, 2794176, 3470040} can be "extended" in that way to 5 new (non primitive) terms (7634088, 18661104, 12448800, 64266048, 38170440).
These have the  q-values (11, 17, 7, 23, 11), so the second and 4th again produce new terms 541172016 and 3020504256 with q-values 29 resp. 47, so both produce new terms 36258525072, 413809083072 with q-values 67, 137, producing 8665787492208, 258630676920000 with q-values 239, 625, producing 11274189527362608, 1101508053002280000 with q-values 1301, 4259, producing 114647233303750360752, 43759610421621577560000 with q-values 10169, 39727, which are again prime, and so it goes on.
The probability that m = A961 <https://emea01.safelinks.protection.outlook.com/?url=https%3A%2F%2Foeis.org%2FA000961&data=04%7C01%7C%7C2fbc68f52f0f4711703008d930579cac%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637593975113222497%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=m6CFIPDs9TQnntC7yQYYEmiHG5RAHvAMn2NqJTtJCi8%3D&reserved=0>(q) is a prime (and then larger than all previous prime factors) is growing as q grows, and in that case we get a new terms.
This is a sufficient, but not necessary condition: e.g., 625 in the above is not prime [but is coprime to 413...72 and therefore produces a new term], but m = A961 <https://emea01.safelinks.protection.outlook.com/?url=https%3A%2F%2Foeis.org%2FA000961&data=04%7C01%7C%7C2fbc68f52f0f4711703008d930579cac%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C637593975113222497%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=m6CFIPDs9TQnntC7yQYYEmiHG5RAHvAMn2NqJTtJCi8%3D&reserved=0>(625) = 4259 is again prime and therefore the chain continues.

So the main difficulty is, as often, not to have "holes" among known terms.

And the main problem is that it is entirely based on oeis.org/A097621 which is actually **plain wrong**!
Indeed, it is defined as
A097621 : In canonical prime factorization of n replace the k-th prime power with k.But, instead of
A246655 Prime powers: numbers of the form p^k where p is a prime and k >= 1.
it uses
A000961 Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1).
If it did use  A246655 as intended in the definition, then A097621 would be not just slightly, but completely different, and as a consequence, even more totally different would be this sequence
oeis.org/A344826 "Integers k such that k/A097621(k) is an integer."
This proves that A097621 and A344826 don't really have a mathematical meaning, they are dependent on the indices arbitrarily assigned to the prime powers.

So the above construction and reasoning may be "interesting" in some sense, and they would still be valid, but would lead to completely different sequences, if the values of A097621 would correspond to its current definition.
(One might "fix" that definition but obviously it does not make any sense to use A961 including 1 in front of the prime powers, when one is only considering the true prime powers (> 1) which appear in factorisations.

- Maximilian

On Mon, Jun 14, 2021 at 6:13 PM Tom Duff <eigenvectors at gmail.com> wrote:
>
> > I think my question wasn't clear enough.
> > Why is this a mathematically interesting sequence?
> > Is the sequence related to some mathematically interesting problem?
> > Does it shed light on some important question?
> > The number of terms in the b file doesn't address those questions.
> >
> > I'm unlikely to be interested in undertaking a long computation if 
> > it doesn't provide some mathematical insight.
> >
> > On Sat, Jun 12, 2021 at 1:44 PM <michel.marcus at free.fr> wrote:
> > >
> > > Well, I see several reasons:
> > > 90 terms for A344826 and A343886 is not much; 2*10^8 is not that 
> > > big
> too.
> > > they are similar to A127724 and so somehow to A127724 (but of 
> > > course,
> > with a different function).
> > > my pari script is limited to the allocatemem I can do, maybe 
> > > someone
> has
> > better.
> > > I tried to save the underlying data in files to counteract my 
> > > memory
> > limitation but when n increase, it kept switching files, so it did 
> > not
> work.

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