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A352052
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Sum of the 6th powers of the divisor complements of the odd proper divisors of n.
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11
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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
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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^6 * Sum_{d|n, d<n, d odd} 1 / d^6.
Sum_{k=1..n} a(k) = c * n^7 / 7, where c = 127*zeta(7)/128 = 1.000471548... . (End)
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EXAMPLE
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a(10) = 10^6 * Sum_{d|10, d<10, d odd} 1 / d^6 = 10^6 * (1/1^6 + 1/5^6) = 1000064.
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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)^6, d = select(`<`, numtheory:-divisors(m), n))
end proc:
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MATHEMATICA
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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 *)
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PROG
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(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
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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), 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).
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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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