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a(n) is the number of k for which C(n,k) is divisible by n.
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%I #29 Sep 20 2024 18:47:40

%S 0,2,1,2,2,4,2,6,4,6,6,10,5,12,6,8,8,16,10,18,10,14,14,22,10,20,18,18,

%T 14,28,11,30,16,26,30,26,22,36,30,30,22,40,20,42,26,26,30,46,20,42,34,

%U 32,34,52,26,46,33,50,42,58,26,60,30,46,32,50,48,66,58,50,44,70,40,72,66,46,58

%N a(n) is the number of k for which C(n,k) is divisible by n.

%C Note that n is prime iff a(n) = n-1. - _T. D. Noe_, Feb 23 2006

%C a(n) >= phi(n) (cf. Robbins). - _Michel Marcus_, Oct 31 2012

%C For n > 0: number of zeros in n-th row of A053200. - _Reinhard Zumkeller_, Jan 01 2013

%H T. D. Noe, <a href="/A020475/b020475.txt">Table of n, a(n) for n = 0..1000</a>

%H H. Harborth, <a href="http://www.jstor.org/stable/2318304">Divisibility of binomial coefficients by their row number</a>, The American Mathematical Monthly, Vol. 84, No. 1 (Jan., 1977), pp. 35-37.

%H N. Robbins, <a href="http://dx.doi.org/10.4153/CMB-1982-052-3">On the number of binomial coefficients which are divisible by their row number</a>, Canad. Math. Bull. 25(1982), 363-365.

%H N. Robbins, <a href="http://dx.doi.org/10.4153/CMB-1985-059-0">On the number of binomial coefficients which are divisible by their row number. II</a>, Canad. Math. Bull. 28(1985), 481-486.

%F a(n) = n + 1 - A007012(n). - _T. D. Noe_, Feb 23 2006

%t Table[cnt=0; Do[If[Mod[Binomial[n,k],n]==0, cnt++ ], {k,0,n}]; cnt,{n,0,100}] (* _T. D. Noe_, Feb 23 2006 *)

%t Join[{0},Table[Count[Table[Binomial[n,k],{k,0,n}],_?(Mod[#,n]==0&)],{n,100}]] (* _Harvey P. Dale_, Sep 20 2024 *)

%o (Haskell)

%o a020475 n = a020475_list !! n

%o a020475_list = 0 : map (sum . map (0 ^)) (tail a053200_tabl)

%o -- _Reinhard Zumkeller_, Jan 24 2014

%Y Cf. A007012.

%K nonn

%O 0,2

%A _David W. Wilson_

%E More terms from _T. D. Noe_, Feb 23 2006