A352050 Sum of the 4th powers of the divisor complements of the odd proper divisors of n.

0, 16, 81, 256, 625, 1312, 2401, 4096, 6642, 10016, 14641, 20992, 28561, 38432, 51331, 65536, 83521, 106288, 130321, 160256, 196963, 234272, 279841, 335872, 391250, 456992, 538083, 614912, 707281, 821312, 923521, 1048576, 1200643, 1336352, 1503651, 1700608, 1874161

Offset

1,2

Links

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

Formula

a(n) = n^4 * Sum_{d|n, d<n, d odd} 1 / d^4.

G.f.: Sum_{k>=2} k^4 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A051001(n) * A006519(n)^4 - A000035(n).

Sum_{k=1..n} a(k) = c * n^5 / 5, where c = 31*zeta(5)/32 = 1.00452376... . (End)

Example

a(10) = 10^4 * Sum_{d|10, d<10, d odd} 1 / d^4 = 10^4 * (1/1^4 + 1/5^4) = 10016.

Maple

f:= proc(n) local m, d;
      m:= n/2^padic:-ordp(n, 2);
      add((n/d)^4, d = select(`<`, numtheory:-divisors(m), n))
end proc:map(f, [$1..40]); # Robert Israel, Apr 03 2023

Mathematica

A352050[n_]:=DivisorSum[n, 1/#^4&, #<n&&OddQ[#]&]n^4; Array[A352050, 50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-4, n/2^IntegerExponent[n, 2]] * n^4 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

Prog

(PARI) a(n) = n^4 * sigma(n >> valuation(n, 2), -4) - n % 2; \\ Amiram Eldar, Oct 13 2023

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), A352049 (k=3), this sequence (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013663, A051001.

Sequence in context: A000583 A050751 A014188 * A050463 A075578 A113316

Adjacent sequences: A352047 A352048 A352049 * A352051 A352052 A352053

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved