A282410 a(n) = binomial(2*p-1, p-1) modulo p^5, where p = prime(n).

3, 10, 126, 1716, 30614, 2198, 1100513, 713337, 4635628, 4511966, 15729649, 49285370, 10820598, 115444165, 110571496, 84562137, 145202954, 386548644, 208729523, 1232287574, 790871562, 2277840181, 3525066856, 4912928962, 7258488370, 8723558568, 9006255935

Offset: 1

Author

Felix Fröhlich, Feb 14 2017

Keywords

• nonn

Comments

Conjecture: a(n) != 1 for all n (cf. McIntosh, 1995, p. 387).

See arXiv:1502.05750, Theorem 2 for several conditions equivalent to p having a(n) = 1.

Clearly, a prime p such that a(n) = 1 must be a Wolstenholme prime, i.e., a term of A088164.

a(n) is prime for n: 1, 7, 19, 59, 76, 92, 109, 112, 165, 196, 221, 249, 263, 326, etc. Robert G. Wilson v, Feb 14 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• C. Aebi and G. Cairns, Wolstenholme again, arXiv:1502.05750 [math.NT], 2015.
• R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica, Vol. 71, No. 4 (1995), 381-389.

Crossrefs

Cf. A088164, A242473.

Programs

Mathematica

f[n_] := Block[{p = Prime@n}, Mod[ Binomial[ 2p -1, p -1], p^5]]; Array[f, 27] (* Robert G. Wilson v, Feb 14 2017 *)
Table[Mod[Binomial[2p-1,p-1],p^5],{p,Prime[Range[30]]}] (* Harvey P. Dale, Jul 07 2022 *)

PARI

a(n) = my(p=prime(n)); lift(Mod(binomial(2*p-1, p-1), p^5))

Python

from sympy import Mod, binomial, prime
def A282410(n): return int(Mod(binomial(2*(p:=prime(n))-1,p-1,evaluate=False),p**5)) # Chai Wah Wu, Apr 24 2025