OFFSET
1,2
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = n^2 * Sum_{d|n, d<n, d odd} 1 / d^2.
G.f.: Sum_{k>=2} k^2 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023
From Amiram Eldar, Oct 13 2023: (Start)
Sum_{k=1..n} a(k) = c * n^3 / 3, where c = 7*zeta(3))/8 = 1.0517997... (A233091). (End)
EXAMPLE
a(10) = 10^2 * Sum_{d|10, d<10, d odd} 1 / d^2 = 10^2 * (1/1^2 + 1/5^2) = 104.
MAPLE
f:= proc(n) local m, d;
m:= n/2^padic:-ordp(n, 2);
add((n/d)^2, d = select(`<`, numtheory:-divisors(m), n))
end proc:
map(f, [$1..60]); # Robert Israel, Apr 03 2023
MATHEMATICA
a[n_] := n^2 DivisorSum[n, If[# < n && OddQ[#], 1/#^2, 0]&];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, May 11 2023 *)
a[n_] := DivisorSigma[-2, n/2^IntegerExponent[n, 2]] * n^2 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)
PROG
(PARI) a(n) = n^2*sumdiv(n, d, if ((d<n) && (d%2), 1/d^2)); \\ Michel Marcus, May 11 2023
(PARI) a(n) = n^2 * sigma(n >> valuation(n, 2), -2) - n % 2; \\ Amiram Eldar, Oct 13 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved