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

1,6

Links

Table of n, a(n) for n=1..105.

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).

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}]

Prog

(PARI) a(n) = sumdiv(n, d, if ((d<n), ispolygonal(d, 3))); \\ Michel Marcus, Apr 14 2021

Crossrefs

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

Sequence in context: A051064 A280509 A153096 * A359236 A320010 A210763

Adjacent sequences: A343404 A343405 A343406 * A343408 A343409 A343410

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 14 2021

Status

approved