A352051 Sum of the 5th powers of the divisor complements of the odd proper divisors of n.

0, 32, 243, 1024, 3125, 7808, 16807, 32768, 59292, 100032, 161051, 249856, 371293, 537856, 762743, 1048576, 1419857, 1897376, 2476099, 3201024, 4101151, 5153664, 6436343, 7995392, 9768750, 11881408, 14408199, 17211392, 20511149, 24407808, 28629151, 33554432, 39296687

Offset

1,2

Links

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

Formula

a(n) = n^5 * Sum_{d|n, d<n, d odd} 1 / d^5.

G.f.: Sum_{k>=2} k^5 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 18 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A051002(n) * A006519(n)^5 - A000035(n).

Sum_{k=1..n} a(k) = c * n^6 / 6, where c = 63*zeta(6)/64 = 1.00144707... . (End)

Example

a(10) = 10^5 * Sum_{d|10, d<10, d odd} 1 / d^5 = 10^5 * (1/1^5 + 1/5^5) = 100032.

Maple

f:= proc(n) local m, d;
      m:= n/2^padic:-ordp(n, 2);
      add((n/d)^5, d = select(`<`, numtheory:-divisors(m), n))
end proc:
map(f, [$1..40]); # Robert Israel, Apr 03 2023

Mathematica

A352051[n_]:=DivisorSum[n, 1/#^5&, #<n&&OddQ[#]&]n^5; Array[A352051, 50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-5, n/2^IntegerExponent[n, 2]] * n^5 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

Prog

(PARI) a(n) = n^5 * sigma(n >> valuation(n, 2), -5) - 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), A352050 (k=4), this sequence (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013664, A051002.

Sequence in context: A257855 A055014 A000584 * A050752 A373000 A351603

Adjacent sequences: A352048 A352049 A352050 * A352052 A352053 A352054

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved