login
A343407
Number of proper divisors of n that are triangular numbers.
1
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
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 14 2021
STATUS
approved