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A352049
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Sum of the cubes of the divisor complements of the odd proper divisors of n.
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11
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = n^3 * Sum_{d|n, d<n, d odd} 1 / d^3.
Sum_{k=1..n} a(k) = c * n^4 / 4, where c = 15*zeta(4)/16 = 1.01467803... (A300707). (End)
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EXAMPLE
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a(10) = 10^3 * Sum_{d|10, d<10, d odd} 1 / d^3 = 10^3 * (1/1^3 + 1/5^3) = 1008.
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MAPLE
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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:
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MATHEMATICA
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a[n_] := DivisorSigma[-3, n/2^IntegerExponent[n, 2]] * n^3 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)
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PROG
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(PARI) a(n) = n^3 * sigma(n >> valuation(n, 2), -3) - n % 2; \\ Amiram Eldar, Oct 13 2023
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CROSSREFS
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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).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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