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%I #22 Nov 17 2017 08:45:24
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,2,1,1,0,2,0,3,0,4,
%T 0,2,0,3,1,2,0,2,0,2,3,5,0,5,0,6,3,3,0,6,1,3,3,8,0,5,0,11,3,8,1,8,0,5,
%U 3,7,0,6,0,8,4,5,1,7,0,7,5,10,0,4,1,9,3,6,0,10,2,8,4,15,2,10,0,16,6,10
%N Cardinality of the set of d, 2<=d<=n/2, which divide binomial(n-d-1,d-1) and are not coprime to n.
%C Note that every d>1 divides binomial(n-d-1,d-1), if gcd(n,d)=1.
%C Numbers n with cardinality 0 are in A138389, with cardinatly 1 in A178071, with cardinality 2 in A178098 and with cardinality 3 in A178099.
%H Antti Karttunen, <a href="/A178101/b178101.txt">Table of n, a(n) for n = 1..16384</a>
%H R. J. Mathar, <a href="http://arxiv.org/abs/1109.0922">Corrigendum to "On the divisibility...."</a>, arXiv:1109.0922 [math.NT], 2011.
%H V. Shevelev, <a href="http://dx.doi.org/10.1142/S179304210700078X">On divisibility of binomial(n-i-1,i-1) by i</a>, Intl. J. of Number Theory, 3, no. 1 (2007), 119-139.
%p A178101 := proc(n) local dvs,d ; dvs := {} ; for d from 1 to n/2 do if gcd(n,d) > 1 and d in numtheory[divisors]( binomial(n-d-1,d-1)) then dvs := dvs union {d} ; end if; end do: nops(dvs) end proc: # _R. J. Mathar_, May 28 2010
%t a[n_] := Sum[Boole[Divisible[Binomial[n-d-1, d-1], d] && !CoprimeQ[d, n]], {d, 2, n/2}];
%t Array[a, 100] (* _Jean-François Alcover_, Nov 17 2017 *)
%o (PARI) a(n) = sum(d=2, n\2, (gcd(d, n) != 1) && ((binomial(n-d-1,d-1) % d) == 0)); \\ _Michel Marcus_, Feb 17 2016
%Y Cf. A138389, A178071, A178098, A179099, A178105.
%K nonn,look
%O 1,26
%A _Vladimir Shevelev_, May 20 2010
%E a(54), a(68), a(70), a(72), a(78) etc corrected by _R. J. Mathar_, May 28 2010
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