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

%I A282410 #17 Apr 25 2025 04:31:19
%S A282410 3,10,126,1716,30614,2198,1100513,713337,4635628,4511966,15729649,
%T A282410 49285370,10820598,115444165,110571496,84562137,145202954,386548644,
%U A282410 208729523,1232287574,790871562,2277840181,3525066856,4912928962,7258488370,8723558568,9006255935
%N A282410 a(n) = binomial(2*p-1, p-1) modulo p^5, where p = prime(n).
%C A282410 Conjecture: a(n) != 1 for all n (cf. McIntosh, 1995, p. 387).
%C A282410 See arXiv:1502.05750, Theorem 2 for several conditions equivalent to p having a(n) = 1.
%C A282410 Clearly, a prime p such that a(n) = 1 must be a Wolstenholme prime, i.e., a term of A088164.
%C A282410 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
%H A282410 Vincenzo Librandi, <a href="/A282410/b282410.txt">Table of n, a(n) for n = 1..1000</a>
%H A282410 C. Aebi and G. Cairns, <a href="https://arxiv.org/abs/1502.05750">Wolstenholme again</a>, arXiv:1502.05750 [math.NT], 2015.
%H A282410 R. J. McIntosh, <a href="https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/71/4/74312">On the converse of Wolstenholme's theorem</a>, Acta Arithmetica, Vol. 71, No. 4 (1995), 381-389.
%t A282410 f[n_] := Block[{p = Prime@n}, Mod[ Binomial[ 2p -1, p -1], p^5]]; Array[f, 27] (* _Robert G. Wilson v_, Feb 14 2017 *)
%t A282410 Table[Mod[Binomial[2p-1,p-1],p^5],{p,Prime[Range[30]]}] (* _Harvey P. Dale_, Jul 07 2022 *)
%o A282410 (PARI) a(n) = my(p=prime(n)); lift(Mod(binomial(2*p-1, p-1), p^5))
%o A282410 (Python)
%o A282410 from sympy import Mod, binomial, prime
%o A282410 def A282410(n): return int(Mod(binomial(2*(p:=prime(n))-1,p-1,evaluate=False),p**5)) # _Chai Wah Wu_, Apr 24 2025
%Y A282410 Cf. A088164, A242473.
%K A282410 nonn
%O A282410 1,1
%A A282410 _Felix Fröhlich_, Feb 14 2017