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A013969
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a(n) = sigma_21(n), the sum of the 21st powers of the divisors of n.
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11
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1, 2097153, 10460353204, 4398048608257, 476837158203126, 21936961102828212, 558545864083284008, 9223376434903384065, 109418989141972712413, 1000000476837160300278, 7400249944258160101212, 46005141850728850805428, 247064529073450392704414, 1171356134499851307229224
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OFFSET
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1,2
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COMMENTS
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If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
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LINKS
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FORMULA
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Sum_{n>=1} a(n)/exp(2*Pi*n) = 77683/552 = Bernoulli(22)/44. - Vaclav Kotesovec, May 07 2023
Multiplicative with a(p^e) = (p^(21*e+21)-1)/(p^21-1).
Dirichlet g.f.: zeta(s)*zeta(s-21).
Sum_{k=1..n} a(k) = zeta(22) * n^22 / 22 + O(n^23). (End)
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MATHEMATICA
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PROG
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(Magma) [DivisorSigma(21, n): n in [1..50]]; // G. C. Greubel, Nov 03 2018
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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