

A179508


a(n) is the unique integer such that sum_{k=0}^{p1}b_k/(n)^k=a(n) (mod p) for any prime p not dividing n, where b_0,b_1,b_2,... are Bell numbers given by A000110.


2



2, 1, 2, 1, 10, 43, 266, 1853, 14834, 133495, 1334962, 14684569, 176214842, 2290792931, 32071101050, 481066515733, 7697064251746, 130850092279663, 2355301661033954, 44750731559645105, 895014631192902122, 18795307255050944539
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OFFSET

1,1


COMMENTS

On July 17, 2010 ZhiWei Sun conjectured that a(n) exists for every n=1,2,3,... He noted that a(1)=2 since sum_{k=0}^{p1}(1)^k*b_k=b_p (mod p), and conjectured that a(2)=1, a(3)=2, a(4)=1, a(5)=10, a(6)=43, a(7)=266, a(8)=1853, a(9)=14834, a(10)=133495. It seems that (1)^{n1}*a(n)>0 for all n=3,4,5,...
I guess that a(2n)=(1)^{n1} (mod 4) and a(2n1)=2 (mod 4) for all n=1,2,3,... Perhaps a(2n1)=2 (mod 8) for every positive integer n. [ZhiWei Sun, Jul 18 2010]
On August 5, 2010 ZhiWei Sun and Don Zagier proved that a(n) actually equals (1)^{n1}D_{n1}+1, where D_0,D_1,D_2,... are derangement numbers given by A000166. [ZhiWei Sun, Aug 07 2010]


LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..451
ZhiWei Sun, Open Conjectures on Congruences, preprint, arXiv:0911.5665 [math.NT], 20092010.
ZhiWei Sun, On Apery numbers and generalized central trinomial coefficients, preprint, arXiv:1006.2776 [math.NT], 20102011.
ZhiWei Sun, A conjecture on Bell numbers (a message to Number Theory List on July 17, 2010) [From ZhiWei Sun, Jul 18 2010]
ZhiWei Sun and Don Zagier, On a curious property of Bell numbers, Bulletin of the Australian Mathematical Society, Volume 84, Issue 1, August 2011. [ZhiWei Sun, Aug 07 2010]


CROSSREFS

Cf. A000110, A000166.
Sequence in context: A248516 A097749 A126906 * A134304 A211096 A134569
Adjacent sequences: A179505 A179506 A179507 * A179509 A179510 A179511


KEYWORD

sign


AUTHOR

ZhiWei Sun, Jul 17 2010


STATUS

approved



