A352052 Sum of the 6th powers of the divisor complements of the odd proper divisors of n.

0, 64, 729, 4096, 15625, 46720, 117649, 262144, 532170, 1000064, 1771561, 2990080, 4826809, 7529600, 11406979, 16777216, 24137569, 34058944, 47045881, 64004096, 85884499, 113379968, 148035889, 191365120, 244156250, 308915840, 387952659, 481894400, 594823321

Offset

1,2

Links

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

Formula

a(n) = n^6 * Sum_{d|n, d<n, d odd} 1 / d^6.

G.f.: Sum_{k>=2} k^6 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 18 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A321810(n) * A006519(n)^6 - A000035(n).

Sum_{k=1..n} a(k) = c * n^7 / 7, where c = 127*zeta(7)/128 = 1.000471548... . (End)

Example

a(10) = 10^6 * Sum_{d|10, d<10, d odd} 1 / d^6 = 10^6 * (1/1^6 + 1/5^6) = 1000064.

Maple

f:= proc(n) local m, d;
      m:= n/2^padic:-ordp(n, 2);
      add((n/d)^6, d = select(`<`, numtheory:-divisors(m), n))
end proc:
map(f, [$1..30]); # Robert Israel, Apr 03 2023

Mathematica

Table[n^6*DivisorSum[n, 1/#^6 &, And[# < n, OddQ[#]] &], {n, 29}] (* Michael De Vlieger, Apr 04 2023 *)
a[n_] := DivisorSigma[-6, n/2^IntegerExponent[n, 2]] * n^6 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

Prog

(PARI) a(n) = n^6*sumdiv(n, d, if ((d<n) && (d%2), 1/d^6)); \\ Michel Marcus, Apr 04 2023
(PARI) a(n) = n^6 * sigma(n >> valuation(n, 2), -6) - 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), A352051 (k=5), this sequence (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A000035, A006519, A013665, A321810.

Sequence in context: A017676 A055015 A001014 * A050753 A074154 A351604

Adjacent sequences: A352049 A352050 A352051 * A352053 A352054 A352055

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved