This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383505 #61 Jul 28 2025 20:18:21
%S A383505 0,1,2,3,4,341,6,79,8,19599,10,3937027727,12,2841,22679,47,16,
%T A383505 18459448019,18,179,146,4003647,22,77934182399,24,299519,80,29952579,
%U A383505 28
%N A383505 Least integer k >= 0 such that binomial(k*n,k+1) = -1 mod n, or -1 if no such integer exists.
%C A383505 Conjecture: a(A000430(n)) = A000430(n)-1.
%C A383505 It is not hard to see that conjecture is true. - _Max Alekseyev_, Jul 24 2025
%C A383505 Other values calculated: a(32) = 1343, a(34) = 1121, a(38) = 417.
%C A383505 If exists, a(30) > 10^13. - _Max Alekseyev_, Jul 24 2025
%H A383505 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts#Binomial_coefficients_modulo_integers">PARI/GP Scripts for Miscellaneous Math Problems: Binomial coefficients modulo integers</a> (binomod.gp).
%F A383505 If p is prime, then a(p) = p-1 by Lucas' theorem. - _Chai Wah Wu_, Jul 21 2025
%e A383505 a(6) = 341 because binomial(341*6, 341+1) = 5 mod 6, and no smaller nonnegative integer satisfies this.
%t A383505 f = {}; Do[k = 0; While[! Mod[Binomial[k*n, k + 1], n] == n - 1, k++]; f = Join[f, {k}], {n, 1, 11}]
%o A383505 (PARI) a(n) = my(k=0); while (binomod(k*n,k+1, n) != Mod(-1, n), k++); k; \\ _Michel Marcus_, May 10 2025
%Y A383505 Cf. A000430, A059288, A099905.
%K A383505 nonn,more
%O A383505 1,3
%A A383505 _Jason Bard_, May 05 2025
%E A383505 a(12) from _Chai Wah Wu_, Jul 21 2025
%E A383505 a(18), a(22), a(24), a(26), a(28) from _Max Alekseyev_, Jul 24 2025