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A352053
Sum of the 7th powers of the divisor complements of the odd proper divisors of n.
11
0, 128, 2187, 16384, 78125, 280064, 823543, 2097152, 4785156, 10000128, 19487171, 35848192, 62748517, 105413632, 170939687, 268435456, 410338673, 612500096, 893871739, 1280016384, 1801914271, 2494358016, 3404825447, 4588568576, 6103593750, 8031810304, 10465138359
OFFSET
1,2
FORMULA
a(n) = n^7 * Sum_{d|n, d<n, d odd} 1 / d^7.
G.f.: Sum_{k>=2} k^7 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 18 2023
From Amiram Eldar, Oct 13 2023: (Start)
a(n) = A321811(n) * A006519(n)^7 - A000035(n).
Sum_{k=1..n} a(k) = c * n^8 / 8, where c = 255*zeta(8)/256 = 1.000155179... . (End)
EXAMPLE
a(10) = 10^7 * Sum_{d|10, d<10, d odd} 1/d^7 = 10^7 * (1/1^7 + 1/5^7) = 10000128.
MATHEMATICA
A352053[n_]:=DivisorSum[n, 1/#^7&, #<n&&OddQ[#]&]n^7; Array[A352053, 50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-7, n/2^IntegerExponent[n, 2]] * n^7 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)
PROG
(PARI) a(n) = n^7 * sigma(n >> valuation(n, 2), -7) - 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), A352052 (k=6), this sequence (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).
Sequence in context: A017678 A123253 A001015 * A050754 A351605 A343287
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved