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

[Olivier Gerard]

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.