A352049 Sum of the cubes of the divisor complements of the odd proper divisors of n.

0, 8, 27, 64, 125, 224, 343, 512, 756, 1008, 1331, 1792, 2197, 2752, 3527, 4096, 4913, 6056, 6859, 8064, 9631, 10656, 12167, 14336, 15750, 17584, 20439, 22016, 24389, 28224, 29791, 32768, 37295, 39312, 43343, 48448, 50653, 54880, 61543, 64512, 68921, 77056, 79507

Offset: 1

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

• nonn
• easy

Examples

			a(10) = 10^3 * Sum_{d|10, d<10, d odd} 1 / d^3 = 10^3 * (1/1^3 + 1/5^3) = 1008.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), this sequence (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A006519, A051000, A300707.

Programs

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&,#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 *)

PARI

a(n) = n^3 * sigma(n >> valuation(n, 2), -3) - n % 2; \\ Amiram Eldar, Oct 13 2023

Formula

a(n) = n^3 * Sum_{d|n, d

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)

a(n) = A051000(n) * A006519(n)^3 - A000035(n).

Sum_{k=1..n} a(k) = c * n^4 / 4, where c = 15*zeta(4)/16 = 1.01467803... (A300707). (End)

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