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A343408
Sum of proper divisors of n that are triangular numbers.
1
0, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 10, 1, 1, 4, 1, 1, 10, 1, 11, 4, 1, 1, 10, 1, 1, 4, 1, 1, 35, 1, 1, 4, 1, 1, 10, 1, 1, 4, 11, 1, 31, 1, 1, 19, 1, 1, 10, 1, 11, 4, 1, 1, 10, 1, 29, 4, 1, 1, 35, 1, 1, 25, 1, 1, 10, 1, 1, 4, 11, 1, 46, 1, 1, 19, 1, 1, 10, 1, 11, 4, 1, 1, 59, 1, 1, 4, 1, 1, 80, 1, 1, 4, 1, 1, 10
OFFSET
1,6
FORMULA
G.f.: Sum_{k>=1} (k*(k+1)/2) * x^(k*(k+1)) / (1 - x^(k*(k+1)/2)).
a(n) = Sum_{d|n, d < n} A010054(d) * d.
MAPLE
a:= n-> add(`if`(issqr(8*d+1), d, 0), d=numtheory[divisors](n) minus {n}):
seq(a(n), n = 1..96); # Alois P. Heinz, Apr 14 2021
MATHEMATICA
nmax = 96; CoefficientList[Series[Sum[(k (k + 1)/2) 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]], d, 0], {d, Divisors[n]}], {n, 96}]
PROG
(PARI) a(n) = sumdiv(n, d, if ((d<n) && ispolygonal(d, 3), d)); \\ Michel Marcus, Apr 14 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 14 2021
STATUS
approved