A372525 Composite numbers k that divide A000296(k) + (-1)^k.

6, 1991, 5459, 17391

Offset: 1

Author

Amiram Eldar, Jun 07 2024

Keywords

• nonn
• hard
• more

Comments

Equivalently, composite numbers k such that Sum_{i=1..k-1} (-1)^i * Bell(i) == 1 (mod k), where Bell(i) = A000110(i).

The congruence holds for all the primes. This is a special case of Sun and Zagier's (2011) Theorem 1.1, when m = 1.

a(5) > 56000, if it exists.

Links

• Grzegorz Serafin, Backward Touchard congruence, Bull. Belg. Math. Soc. Simon Stevin, Vol. 28, No. 4 (2022), pp. 603-614; arXiv preprint, arXiv:2110.06129 [math.NT], 2021.
• Zhi-Wei Sun and Don Zagier, On a curious property of Bell numbers, Bulletin of the Australian Mathematical Society, Vol. 84, No. 1 (2011), pp. 153-158.

Crossrefs

Cf. A000110, A000296, A179508.

Programs

Mathematica

f[k_] := f[k] = Sum[Binomial[k-1, i] * f[k-i-1], {i, 1, k-1}]; f[0] = 1; Select[Range[2000], CompositeQ[#] && Divisible[f[#] + (-1)^# , #] &]