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A317872 a(n) is the number of times that binomial(n+m, m) mod m = 1, for 0 < m <= n.

%I A317872 #25 Aug 18 2022 18:11:37
%S A317872 0,0,0,1,1,0,0,1,3,2,2,1,1,0,0,2,2,1,1,3,2,0,0,1,2,2,4,4,4,0,0,3,4,1,
%T A317872 1,3,3,1,0,1,1,2,2,1,2,1,1,2,4,2,2,4,4,6,4,3,2,2,2,1,1,0,1,7,6,1,1,1,
%U A317872 1,1,1,3,3,2,2,2,2,1,1,3,7,4,4,4,4,0,1,2,2,2,2,2,1,0,0,3,3,3,4,5,5,2,2,3,1
%N A317872 a(n) is the number of times that binomial(n+m, m) mod m = 1, for 0 < m <= n.
%C A317872 Inspired by A133906.
%C A317872 First occurrence of k with k = 0, 1, 2,...: 1, 4, 10, 9, 27, 100, 54, 64, 176, 544, 352, 648, 649, 129, 128, 1378, 513, 729, 7776, 1377, 5832, 1701, 3728, 13312, 13825, ...
%C A317872 Records: 0, 1, 3, 4, 6, 7, 14, 16, 17, 19, 21, 22, ..., ; and they occur at: 1, 4, 9, 27, 54, 64, 128, 513, 729, 1377, 1701, 3728, 6656, ...
%H A317872 Hieronymus Fischer, <a href="/A317872/b317872.txt">Table of n, a(n) for n = 1..10000</a>
%e A317872 a(9) = 3 because binomial(9+2,2) mod 2 = binomial(9+3,3) mod 3 = binomial(9+6,6) mod 6 = 1. [Corrected by _Jon E. Schoenfield_, Aug 28 2018]
%t A317872 a[n_] := Block[{c = 0, m = 1}, While[m < n + 1, If[ Mod[ Binomial[n + m, m], m] == 1, c++]; m++]; c]; Array[a, 105]
%t A317872 Table[Count[Table[Mod[Binomial[n+m,m],m],{m,n}],1],{n,120}] (* _Harvey P. Dale_, Aug 18 2022 *)
%Y A317872 Cf. A133906.
%K A317872 nonn
%O A317872 1,9
%A A317872 _Hieronymus Fischer_ and _Robert G. Wilson v_, Aug 09 2018