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A013966
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a(n) = sigma_18(n), the sum of the 18th powers of the divisors of n.
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6
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1, 262145, 387420490, 68719738881, 3814697265626, 101560344351050, 1628413597910450, 18014467229220865, 150094635684419611, 1000003814697527770, 5559917313492231482, 26623434909949071690, 112455406951957393130, 426880482624234915250, 1477891883850485076740
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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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Multiplicative with a(p^e) = (p^(18*e+18)-1)/(p^18-1).
Dirichlet g.f.: zeta(s)*zeta(s-18).
Sum_{k=1..n} a(k) = zeta(19) * n^19 / 19. + O(n^20). (End)
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MATHEMATICA
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PROG
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(Magma) [DivisorSigma(18, 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,mult,easy
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AUTHOR
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STATUS
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approved
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