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A385558 Period of {binomial(N,7) mod n: N in Z}.

%I A385558 #18 Jul 07 2025 11:54:12
%S A385558 1,8,9,16,25,72,49,32,27,200,11,144,13,392,225,64,17,216,19,400,441,
%T A385558 88,23,288,125,104,81,784,29,1800,31,128,99,136,1225,432,37,152,117,
%U A385558 800,41,3528,43,176,675,184,47,576,343,1000,153,208,53,648,275,1568,171,232,59,3600
%N A385558 Period of {binomial(N,7) mod n: N in Z}.
%H A385558 Jianing Song, <a href="/A385558/b385558.txt">Table of n, a(n) for n = 1..10000</a>
%F A385558 Multiplicative with a(2^e) = 2^(e+2), a(3^e) = 3^(e+1), a(5^e) = 5^(e+1), a(7^e) = 7^(e+1), and a(p^e) = p^e for primes p >= 11.
%F A385558 From _Amiram Eldar_, Jul 07 2025: (Start)
%F A385558 a(n) = n * gcd(210, n) * (2 - (n mod 2)).
%F A385558 Dirichlet g.f.: zeta(s-1) * (1 + 3/2*(s-1)) * (1 + 2/3*(s-1)) * (1 + 4/5*(s-1)) * (1 + 6/7*(s-1)).
%F A385558 Sum_{k=1..n} a(k) ~ (195/28) * n^2. (End)
%e A385558 For N == 0, 1, ..., 48 (mod 49), binomial(N,7) == {0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6} (mod 7).
%t A385558 A385558[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 7]], {p, FactorInteger[n][[All, 1]]}]];
%t A385558 Array[A385558, 100] (* _Paolo Xausa_, Jul 07 2025 *)
%t A385558 a[n_] := n * GCD[n, 210] * (2 - Mod[n, 2]); Array[a, 100] (* _Amiram Eldar_, Jul 07 2025 *)
%o A385558 (PARI) a(n, {choices=7}) = 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 A385558 Row n = 7 of A349593. A022998, A385555, A385556, A385557, A385559, and A385560 are respectively rows 2, 3, 4, 5-6, 8, and 9-10.
%K A385558 nonn,easy,mult
%O A385558 1,2
%A A385558 _Jianing Song_, Jul 03 2025