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A179508 a(n) is the unique integer such that sum_{k=0}^{p-1}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 (list; graph; refs; listen; history; text; internal format)



On July 17, 2010 Zhi-Wei Sun conjectured that a(n) exists for every n=1,2,3,... He noted that a(1)=2 since sum_{k=0}^{p-1}(-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)^{n-1}*a(n)>0 for all n=3,4,5,...

I guess that a(2n)=(-1)^{n-1} (mod 4) and a(2n-1)=2 (mod 4) for all n=1,2,3,... Perhaps a(2n-1)=2 (mod 8) for every positive integer n. [Zhi-Wei Sun, Jul 18 2010]

On August 5, 2010 Zhi-Wei Sun and Don Zagier proved that a(n) actually equals (-1)^{n-1}D_{n-1}+1, where D_0,D_1,D_2,... are derangement numbers given by A000166. [Zhi-Wei Sun, Aug 07 2010]


Seiichi Manyama, Table of n, a(n) for n = 1..451

Zhi-Wei Sun, Open Conjectures on Congruences, preprint, arXiv:0911.5665 [math.NT], 2009-2010.

Zhi-Wei Sun, On Apery numbers and generalized central trinomial coefficients, preprint, arXiv:1006.2776 [math.NT], 2010-2011.

Zhi-Wei Sun, A conjecture on Bell numbers (a message to Number Theory List on July 17, 2010) [From Zhi-Wei Sun, Jul 18 2010]

Zhi-Wei Sun and Don Zagier, On a curious property of Bell numbers, Bulletin of the Australian Mathematical Society, Volume 84, Issue 1, August 2011. [Zhi-Wei Sun, Aug 07 2010]


Cf. A000110, A000166.

Sequence in context: A248516 A097749 A126906 * A134304 A211096 A134569

Adjacent sequences:  A179505 A179506 A179507 * A179509 A179510 A179511




Zhi-Wei Sun, Jul 17 2010



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Last modified August 11 15:12 EDT 2020. Contains 336428 sequences. (Running on oeis4.)