OFFSET
1,2
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = n^3 * Sum_{d|n, d<n, d odd} 1 / d^3.
G.f.: Sum_{k>=2} k^3 * 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^4 / 4, where c = 15*zeta(4)/16 = 1.01467803... (A300707). (End)
EXAMPLE
a(10) = 10^3 * Sum_{d|10, d<10, d odd} 1 / d^3 = 10^3 * (1/1^3 + 1/5^3) = 1008.
MAPLE
f:= proc(n) local m, d;
m:= n/2^padic:-ordp(n, 2);
add((n/d)^3, d = select(`<`, numtheory:-divisors(m), n))
end proc:
map(f, [$1..50]); # Robert Israel, Apr 03 2023
MATHEMATICA
A352049[n_]:=DivisorSum[n, 1/#^3&, #<n&&OddQ[#]&]n^3; Array[A352049, 50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-3, n/2^IntegerExponent[n, 2]] * n^3 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)
PROG
(PARI) a(n) = n^3 * sigma(n >> valuation(n, 2), -3) - n % 2; \\ Amiram Eldar, Oct 13 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved