cp's OEIS Frontend

A343407 Number of proper divisors of n that are triangular numbers.

%I A343407 #9 Apr 14 2021 09:20:34
%S A343407 0,1,1,1,1,2,1,1,2,1,1,3,1,1,2,1,1,3,1,2,2,1,1,3,1,1,2,1,1,5,1,1,2,1,
%T A343407 1,3,1,1,2,2,1,4,1,1,3,1,1,3,1,2,2,1,1,3,1,2,2,1,1,5,1,1,3,1,1,3,1,1,
%U A343407 2,2,1,4,1,1,3,1,1,3,1,2,2,1,1,5,1,1,2,1,1,6,1,1,2,1,1,3,1,1,2,2,1,3,1,1,4
%N A343407 Number of proper divisors of n that are triangular numbers.
%F A343407 G.f.: Sum_{k>=1} x^(k*(k+1)) / (1 - x^(k*(k+1)/2)).
%F A343407 a(n) = Sum_{d|n, d < n} A010054(d).
%p A343407 a:= n-> add(`if`(issqr(8*d+1), 1, 0), d=numtheory[divisors](n) minus {n}):
%p A343407 seq(a(n), n = 1..105);  # _Alois P. Heinz_, Apr 14 2021
%t A343407 nmax = 105; CoefficientList[Series[Sum[x^(k (k + 1))/(1 - x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A343407 Table[Sum[If[d < n && IntegerQ[Sqrt[8 d + 1]], 1, 0], {d, Divisors[n]}], {n, 105}]
%o A343407 (PARI) a(n) = sumdiv(n, d, if ((d<n), ispolygonal(d, 3))); \\ _Michel Marcus_, Apr 14 2021
%Y A343407 Cf. A000217, A007862, A010054, A032741, A293234, A343408.
%K A343407 nonn
%O A343407 1,6
%A A343407 _Ilya Gutkovskiy_, Apr 14 2021