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A385556 Period of {binomial(N,4) mod n: N in Z}.

%I A385556 #19 Jul 07 2025 11:50:34
%S A385556 1,8,9,16,5,72,7,32,27,40,11,144,13,56,45,64,17,216,19,80,63,88,23,
%T A385556 288,25,104,81,112,29,360,31,128,99,136,35,432,37,152,117,160,41,504,
%U A385556 43,176,135,184,47,576,49,200,153,208,53,648,55,224,171,232,59,720
%N A385556 Period of {binomial(N,4) mod n: N in Z}.
%H A385556 Jianing Song, <a href="/A385556/b385556.txt">Table of n, a(n) for n = 1..10000</a>
%F A385556 Multiplicative with a(2^e) = 2^(e+2), a(3^e) = 3^(e+1), and a(p^e) = p^e for primes p >= 5.
%F A385556 From _Amiram Eldar_, Jul 07 2025: (Start)
%F A385556 a(n) = n * gcd(6, n) * (2 - (n mod 2)) = n * A089128(n) * A000034(n-1).
%F A385556 Dirichlet g.f.: zeta(s-1) * (1 + 3/2*(s-1)) * (1 + 2/3*(s-1)).
%F A385556 Sum_{k=1..n} a(k) ~ (25/12) * n^2. (End)
%e A385556 For N == 0, 1, ..., 31 (mod 32), binomial(N,4) == {0, 0, 0, 0, 1, 5, 7, 3, 6, 6, 2, 2, 7, 3, 1, 5, 4, 4, 4, 4, 5, 1, 3, 7, 2, 2, 6, 6, 3, 7, 5, 1} (mod 8).
%t A385556 A385556[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 4]], {p, FactorInteger[n][[All, 1]]}]];
%t A385556 Array[A385556, 100] (* _Paolo Xausa_, Jul 07 2025 *)
%t A385556 a[n_] := n * GCD[n, 6] * (2 - Mod[n, 2]); Array[a, 100] (* _Amiram Eldar_, Jul 07 2025 *)
%o A385556 (PARI) a(n, {choices=4}) = 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 A385556 Row n = 4 of A349593. A022998, A385555, A385557, A385558, A385559, and A385560 are respectively rows n = 2, 3, 5-6, 7, 8, and 9-10.
%Y A385556 Cf. A000034, A089128.
%K A385556 nonn,easy,mult
%O A385556 1,2
%A A385556 _Jianing Song_, Jul 03 2025