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A385557 Period of {binomial(N,5) mod n: N in Z}. Also, period of {binomial(N,6) mod n: N in Z}.

%I A385557 #19 Jul 07 2025 11:54:26
%S A385557 1,8,9,16,25,72,7,32,27,200,11,144,13,56,225,64,17,216,19,400,63,88,
%T A385557 23,288,125,104,81,112,29,1800,31,128,99,136,175,432,37,152,117,800,
%U A385557 41,504,43,176,675,184,47,576,49,1000,153,208,53,648,275,224,171,232,59,3600
%N A385557 Period of {binomial(N,5) mod n: N in Z}. Also, period of {binomial(N,6) mod n: N in Z}.
%H A385557 Jianing Song, <a href="/A385557/b385557.txt">Table of n, a(n) for n = 1..10000</a>
%F A385557 Multiplicative with a(2^e) = 2^(e+2), a(3^e) = 3^(e+1), a(5^e) = 5^(e+1), and a(p^e) = p^e for primes p >= 7.
%F A385557 From _Amiram Eldar_, Jul 07 2025: (Start)
%F A385557 a(n) = n * gcd(30, n) * (2 - (n mod 2)).
%F A385557 Dirichlet g.f.: zeta(s-1) * (1 + 3/2*(s-1)) * (1 + 2/3*(s-1)) * (1 + 4/5*(s-1)).
%F A385557 Sum_{k=1..n} a(k) ~ (15/4) * n^2. (End)
%e A385557 For N == 0, 1, ..., 71 (mod 72), binomial(N,5) == {0, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 0, 0, 3, 4, 3, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 2, 0, 0, 3, 0, 3, 4, 0, 0, 2, 0, 3, 0, 3, 0, 4, 0, 0, 2, 3, 0, 3, 0, 0, 4, 0, 0, 5, 0, 3, 0, 0, 0, 4, 0, 3, 2, 3, 0, 0, 0, 0, 4, 3, 0, 5} (mod 6), and binomial(N,6) == {0, 0, 0, 0, 0, 0, 1, 1, 4, 0, 0, 0, 0, 0, 3, 1, 4, 4, 0, 0, 0, 0, 3, 3, 4, 4, 4, 0, 0, 0, 3, 3, 0, 4, 4, 4, 0, 0, 3, 3, 0, 0, 4, 4, 4, 0, 3, 3, 0, 0, 0, 4, 4, 4, 3, 3, 0, 0, 0, 0, 4, 4, 1, 3, 0, 0, 0, 0, 0, 4, 1, 1} (mod 6).
%t A385557 A385557[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 5]], {p, FactorInteger[n][[All, 1]]}]];
%t A385557 Array[A385557, 100] (* _Paolo Xausa_, Jul 07 2025 *)
%t A385557 a[n_] := n * GCD[n, 30] * (2 - Mod[n, 2]); Array[a, 100] (* _Amiram Eldar_, Jul 07 2025 *)
%o A385557 (PARI) a(n, {choices=5}) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); r *= p^(logint(choices, p)+e)); return(r)
%Y A385557 Rows n = 5 and 6 of A349593. A022998, A385555, A385556, A385558, A385559, and A385560 are respectively rows 2, 3, 4, 7, 8, and 9-10.
%K A385557 nonn,easy,mult
%O A385557 1,2
%A A385557 _Jianing Song_, Jul 03 2025