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A385555 Period of {binomial(N,3) mod n: N in Z}.

%I A385555 #18 Jul 07 2025 11:49:23
%S A385555 1,4,9,8,5,36,7,16,27,20,11,72,13,28,45,32,17,108,19,40,63,44,23,144,
%T A385555 25,52,81,56,29,180,31,64,99,68,35,216,37,76,117,80,41,252,43,88,135,
%U A385555 92,47,288,49,100,153,104,53,324,55,112,171,116,59,360
%N A385555 Period of {binomial(N,3) mod n: N in Z}.
%H A385555 Jianing Song, <a href="/A385555/b385555.txt">Table of n, a(n) for n = 1..10000</a>
%F A385555 Multiplicative with a(2^e) = 2^(e+1), a(3^e) = 3^(e+1), and a(p^e) = p^e for primes p >= 5.
%F A385555 From _Amiram Eldar_, Jul 07 2025: (Start)
%F A385555 a(n) = n * gcd(6, n) = n * A089128(n).
%F A385555 Dirichlet g.f.: zeta(s-1) * (1 + 1/2*(s-1)) * (1 + 2/3*(s-1)).
%F A385555 Sum_{k=1..n} a(k) ~ (5/4) * n^2. (End)
%e A385555 For N == 0, 1, ..., 26 (mod 27), binomial(N,3) == {0, 0, 0, 1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8} (mod 4).
%t A385555 A385555[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 3]], {p, FactorInteger[n][[All, 1]]}]];
%t A385555 Array[A385555, 100] (* _Paolo Xausa_, Jul 07 2025 *)
%t A385555 a[n_] := n * GCD[n, 6]; Array[a, 100] (* _Amiram Eldar_, Jul 07 2025 *)
%o A385555 (PARI) a(n, {choices=3}) = 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 A385555 Row n = 3 of A349593. A022998, A385556, A385557, A385558, A385559, and A385560 are respectively rows n = 2, 4, 5-6, 7, 8, and 9-10.
%Y A385555 Cf. A089128.
%K A385555 nonn,easy,mult
%O A385555 1,2
%A A385555 _Jianing Song_, Jul 03 2025