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

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, 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, 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

Offset: 1

Author

Ilya Gutkovskiy, Apr 14 2021

Keywords

• nonn

Crossrefs

Cf. A000217, A007862, A010054, A032741, A293234, A343408.

Programs

Maple

a:= n-> add(`if`(issqr(8*d+1), 1, 0), d=numtheory[divisors](n) minus {n}):
seq(a(n), n = 1..105);  # Alois P. Heinz, Apr 14 2021

Mathematica

nmax = 105; CoefficientList[Series[Sum[x^(k (k + 1))/(1 - x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[If[d < n && IntegerQ[Sqrt[8 d + 1]], 1, 0], {d, Divisors[n]}], {n, 105}]

PARI

a(n) = sumdiv(n, d, if ((dMichel Marcus, Apr 14 2021

Formula

G.f.: Sum_{k>=1} x^(k*(k+1)) / (1 - x^(k*(k+1)/2)).

a(n) = Sum_{d|n, d < n} A010054(d).