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A222760 Conjectured least prime p for which binomial(n*q,q) (mod q^3) = n for all primes q >= p.

%I A222760 #9 Feb 16 2025 08:33:19
%S A222760 2,5,3,2,5,3,3,5,3,3,5,3,3,5,3,3,5,3,3,5,3,3,5,3,3,5,5,5,5,5,5,5,5,5,
%T A222760 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
%U A222760 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A222760 Conjectured least prime p for which binomial(n*q,q) (mod q^3) = n for all primes q >= p.
%C A222760 The n = 2 case is mentioned in Eric Weisstein's website.
%H A222760 T. D. Noe, <a href="/A222760/b222760.txt">Table of n, a(n) for n = 1..10000</a>
%H A222760 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/CentralBinomialCoefficient.html">MathWorld: Central Binomial Coefficient</a>
%t A222760 lim = 100; Table[r = Table[Mod[Binomial[n*p, p], p^3] == n, {p, Prime[Range[lim]]}]; i = lim; While[i > 0 && r[[i]], i--]; Prime[i + 1], {n, 87}]
%Y A222760 Cf. A096328 (prime(prime(n)^3)), A222759.
%K A222760 nonn
%O A222760 1,1
%A A222760 _T. D. Noe_, Mar 13 2013