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A278963 a(n) is the number of k, 1<=k<=n, such that gcd(n,k) divides binomial(n,k).

%I A278963 #11 Apr 29 2019 08:24:33
%S A278963 1,1,2,3,4,2,6,6,8,6,10,7,12,6,8,14,16,14,18,12,14,14,22,12,24,18,24,
%T A278963 18,28,11,30,28,26,30,26,28,36,30,30,27,40,20,42,30,32,30,46,32,48,42,
%U A278963 32,38,52,36,46,43,50,42,58,32,60,30,52,60,50,48,66,60,50
%N A278963 a(n) is the number of k, 1<=k<=n, such that gcd(n,k) divides binomial(n,k).
%C A278963 Length of row n of A278961.
%C A278963 a(n) is odd if and only if n = 1 or n is in A048618.
%C A278963 a(n) <= n-1 for n>1.
%C A278963 a(n) = n-1 if n is a prime or the square of a prime.
%C A278963 a(n) >= A000010(n).
%H A278963 Robert Israel, <a href="/A278963/b278963.txt">Table of n, a(n) for n = 1..10000</a>
%e A278963 a(8) = 6 because gcd(8,k) divides binomial(8,k) for k=1,2,3,5,6,7 but not k=4 or k=8.
%p A278963 f:= proc(n,m) if binomial(n,m) mod igcd(n,m) = 0 then m else NULL fi end proc:
%p A278963 [seq(nops([seq(f(n,m),m=1..n)]),n=1..200)];
%t A278963 a[n_] := Sum[Boole[Divisible[Binomial[n, k], GCD[n, k]]], {k, 1, n}];
%t A278963 Array[a, 100] (* _Jean-François Alcover_, Apr 29 2019 *)
%o A278963 (PARI) a(n) = sum(k=1, n, (binomial(n, k) % gcd(n, k))==0); \\ _Michel Marcus_, Dec 04 2016
%Y A278963 Cf. A000010, A048618, A278961.
%K A278963 nonn
%O A278963 1,3
%A A278963 _Robert Israel_, Dec 02 2016