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A352032
Sum of the 4th powers of the odd proper divisors of n.
11
0, 1, 1, 1, 1, 82, 1, 1, 82, 626, 1, 82, 1, 2402, 707, 1, 1, 6643, 1, 626, 2483, 14642, 1, 82, 626, 28562, 6643, 2402, 1, 51332, 1, 1, 14723, 83522, 3027, 6643, 1, 130322, 28643, 626, 1, 196964, 1, 14642, 57893, 279842, 1, 82, 2402, 391251, 83603, 28562, 1, 538084, 15267
OFFSET
1,6
FORMULA
a(n) = Sum_{d|n, d<n, d odd} d^4.
G.f.: Sum_{k>=1} (2*k-1)^4 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
From Amiram Eldar, Oct 11 2023: (Start)
a(n) = A051001(n) - n^4*A000035(n).
Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)-1)/10 = 0.0036927755... . (End)
EXAMPLE
a(10) = 626; a(10) = Sum_{d|10, d<10, d odd} d^4 = 1^4 + 5^4 = 626.
MATHEMATICA
f[2, e_] := 1; f[p_, e_] := (p^(4*e+4) - 1)/(p^4 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^4, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)
CROSSREFS
Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), this sequence (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
Sequence in context: A340398 A347172 A347175 * A051001 A363991 A050678
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved