[seqfan] Re: Richard Guy: old aliquot and divisibility sequences (by way of list admin)

[Neil Sloane]

Olivier, I entered it myself the other day: see A214525 - right?
There was a mistake in one of Richard's terms.
Neil

On Sat, Aug 11, 2012 at 10:18 AM, Olivier Gerard
<olivier.gerard at gmail.com>wrote:

> Hello,
>
> Just going through old email, and it seems that this sequence is
> still not in OEIS.  Would someone do the necessary?
>
> Thanks!  R.
>
>
> ---------- Forwarded message ----------
> Date: Wed, 29 Jul 2009 10:05:53 -0600 (MDT)
> From: Richard Guy <rkg at cpsc.ucalgary.ca>
> Reply-To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> To: Sequence Fans <seqfan at list.seqfan.eu>
> Subject: [seqfan]  New sequence?
>
> Apologies that this is not in the approved format.
> Perhaps a kind soul will do the necessary.  The
> factorization of many sequences is of interest,
> e.g., divisibility sequences, aliquot sequences,
> but it may be too clumsy to display these.  The
> following is a divisibility sequence.  Details
> are given below.  A check is advisable.   R.
>
> 0, 1, 7, 19, 21 = 3.7, 4 = 2^2, 133 = 7.19, 937,
> 2667 = 3.7.127, 3429 = 19.181, 2128 = 2^4.7.19,
> 20569 = 67.307, 132867 = 3^3.7.19.37,
> 392743 = 13.30211, 596869 = 7^2.13.937,
> 647596 = 2^2.19.8521, 3539109 = 3.7.127.1327,
> 19881229 = p, 60254719 = 7.19.181.2503,
> 106198903 = 1597.66499, 158297664 = 2^6.3.7.19.6100,
> 643809889 = 19.29^2.43.937, 3117087967 = 7.67.307.21649
> 9564827611 = 1609.5944579,
> 19050869061 = 3^3.7.19.37.127.1129,
> 34555674196 = 2^2.199^2.218149,
> 119658973525 = 5^2.7.13.1741.30211,
> 507648339217 = 19.109.181.1354267,
> 1561117435059 = 3.7^2.13.937.871837,
> 3421971910543 = 11833.289188871,
> 7059581286352 = 2^4.7.19^2.31.661.8521,
> 22331700758233 = 61.269389.1358977,
> 85133405020251 = 3.7.31.127.673.1153.1327,
> 260547577117039 = 19.67.307.463.991.1453,
> 614098578475669 = 7.103.42841.19881229,
> 1383379284476668 = 2^2.937.3499.105486709,
> 4154682119138901 = 3^5.7.19.37.181.2503.7669,
> 14589050357581813 = 73.5623.35541526747,
> 44260418356926919 = 7.37.1597.66499.1609147.
> 110068609172556151 = 13.19.157.547.30211.171757,
> 263774685682276608 = 2^8.3.7.19.79.127.6199.41521,
> 768741820775054977 = 104959.7324210603903, ...
>
> It's a 4th order recurrence, with relation
>
> a(n) = 7a(n-1) - 23a(n-2) + 49a(n-3) - 49a(n-4)
>
> It factors over the Eisenstein-Jacobi integers
> into two 2nd order sequences  (w^3 = 1)
>
> 0, 1, w+3, 3w+5, 4w+5, 2, -12w-1, -29w+3, ...
>
> and its conjugate (replace  w  by  w^2).
>
> The relation for this is
>
>        a(n) = (w+3)a(n-1) - (2w+3)a(n-2).
>
> I recently noticed (and it may not be widely
> known) that the primes in 4th (16th, etc) order
> sequences may have 2 (4, etc) ranks of apparition.
> A paper on this by Hugh Williams, Matt Greenberg
> and me may appear some day.  Here
>
> p = 13  has ranks of apparition  13 & 14,
> p = 19   ...   ...   ...   ...    3 & 10,
> p = 31   ...   ...   ...   ...   30 & 32,
> etc.
>
> In this case the ranks are divisors of
> p - 1  and  p + 1.   Exception  p = 13,
> 'cos  13  divides the discriminant, -3.7^2.13^2
>
> It's important to notice that  2 + 2 != 2 x 2.
> If you multiply the quadratics
>
> x^2 - (w+3)x + (2w+3) and x^2 - (w^2+3)x + (2w^2+3)
>
> you don't get the biquadratic
>
>         x^4 - 7x^3 + 23x^2 - 49x + 49
>
> The correct relationship is left to the reader.
>
> I have infinitely many more, if anyone is
> interested.  With next to no encouragement
> I'll send a 16th order one.       R.
>
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>
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>



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Neil J. A. Sloane, President, OEIS Foundation
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