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A385560 Period of {binomial(N,9) mod n: N in Z}. Also, period of {binomial(N,10) mod n: N in Z}.

%I A385560 #19 Jul 07 2025 11:54:50
%S A385560 1,16,27,32,25,432,49,64,81,400,11,864,13,784,675,128,17,1296,19,800,
%T A385560 1323,176,23,1728,125,208,243,1568,29,10800,31,256,297,272,1225,2592,
%U A385560 37,304,351,1600,41,21168,43,352,2025,368,47,3456,343,2000,459,416,53,3888,275,3136,513,464,59,21600
%N A385560 Period of {binomial(N,9) mod n: N in Z}. Also, period of {binomial(N,10) mod n: N in Z}.
%H A385560 Jianing Song, <a href="/A385560/b385560.txt">Table of n, a(n) for n = 1..10000</a>
%F A385560 Multiplicative with a(2^e) = 2^(e+3), a(3^e) = 3^(e+2), a(5^e) = 5^(e+1), a(7^e) = 7^(e+1), and a(p^e) = p^e for primes p >= 11.
%F A385560 From _Amiram Eldar_, Jul 07 2025: (Start)
%F A385560 a(n) = n * gcd(6, n) * gcd(210, n) * (2 - (n mod 2)).
%F A385560 Dirichlet g.f.: zeta(s-1) * (1 + 7/2*(s-1)) * (1 + 8/3*(s-1)) * (1 + 4/5*(s-1)) * (1 + 6/7*(s-1)).
%F A385560 Sum_{k=1..n} a(k) ~ (3861/140) * n^2. (End)
%e A385560 For N == 0, 1, ..., 80 (mod 81), binomial(N,9) == {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 4, 4, 4, 1, 1, 1, 2, 2, 2, 8, 8, 8, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 7, 7, 7, 4, 4, 4, 5, 5, 5, 2, 2, 2, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 1, 1, 1, 7, 7, 7, 8, 8, 8, 5, 5, 5, 8, 8, 8} (mod 9), and binomial(N,10) == {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 7, 2, 6, 7, 8, 0, 2, 4, 6, 5, 4, 3, 5, 7, 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 4, 8, 3, 1, 8, 6, 1, 5, 0, 5, 1, 6, 8, 1, 3, 8, 4, 0, 6, 3, 0, 6, 3, 0, 6, 3, 0, 7, 5, 3, 4, 5, 6, 4, 2, 0, 8, 7, 6, 2, 7, 3, 2, 1} (mod 9).
%t A385560 A385560[n_] := If[n == 1, 1, n*Product[p^Floor[Log[p, 9]], {p, FactorInteger[n][[All, 1]]}]];
%t A385560 Array[A385560, 100] (* _Paolo Xausa_, Jul 07 2025 *)
%t A385560 a[n_] := n * GCD[n, 6] * GCD[n, 210] * (2 - Mod[n, 2]); Array[a, 100] (* _Amiram Eldar_, Jul 07 2025 *)
%o A385560 (PARI) a(n, {choices=10}) = 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 A385560 Rows n = 9 and 10 of A349593. A022998, A385555, A385556, A385557, A385558, and A385559 are respectively rows 2, 3, 4, 5-6, 7, and 8.
%K A385560 nonn,easy,mult
%O A385560 1,2
%A A385560 _Jianing Song_, Jul 03 2025