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A352049
Sum of the cubes of the divisor complements of the odd proper divisors of n.
11
0, 8, 27, 64, 125, 224, 343, 512, 756, 1008, 1331, 1792, 2197, 2752, 3527, 4096, 4913, 6056, 6859, 8064, 9631, 10656, 12167, 14336, 15750, 17584, 20439, 22016, 24389, 28224, 29791, 32768, 37295, 39312, 43343, 48448, 50653, 54880, 61543, 64512, 68921, 77056, 79507
OFFSET
1,2
LINKS
FORMULA
a(n) = n^3 * Sum_{d|n, d<n, d odd} 1 / d^3.
G.f.: Sum_{k>=2} k^3 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023
From Amiram Eldar, Oct 13 2023: (Start)
a(n) = A051000(n) * A006519(n)^3 - A000035(n).
Sum_{k=1..n} a(k) = c * n^4 / 4, where c = 15*zeta(4)/16 = 1.01467803... (A300707). (End)
EXAMPLE
a(10) = 10^3 * Sum_{d|10, d<10, d odd} 1 / d^3 = 10^3 * (1/1^3 + 1/5^3) = 1008.
MAPLE
f:= proc(n) local m, d;
m:= n/2^padic:-ordp(n, 2);
add((n/d)^3, d = select(`<`, numtheory:-divisors(m), n))
end proc:
map(f, [$1..50]); # Robert Israel, Apr 03 2023
MATHEMATICA
A352049[n_]:=DivisorSum[n, 1/#^3&, #<n&&OddQ[#]&]n^3; Array[A352049, 50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-3, n/2^IntegerExponent[n, 2]] * n^3 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)
PROG
(PARI) a(n) = n^3 * sigma(n >> valuation(n, 2), -3) - 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), this sequence (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).
Sequence in context: A125084 A052048 A052064 * A125496 A030289 A111131
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved