[seqfan] New sequence?
Richard Guy
rkg at cpsc.ucalgary.ca
Wed Jul 29 18:05:53 CEST 2009
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.
More information about the SeqFan
mailing list