OFFSET
1,1
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.
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)^# , #] &]
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Amiram Eldar, Jun 07 2024
STATUS
approved