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A023887
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a(n) = sigma_n(n): sum of n-th powers of divisors of n.
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56
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1, 5, 28, 273, 3126, 47450, 823544, 16843009, 387440173, 10009766650, 285311670612, 8918294543346, 302875106592254, 11112685048647250, 437893920912786408, 18447025552981295105, 827240261886336764178, 39346558271492178925595, 1978419655660313589123980
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OFFSET
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1,2
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COMMENTS
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Compare to A217872(n) = sigma(n)^n.
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REFERENCES
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Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
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LINKS
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FORMULA
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If the canonical prime factorization of n > 1 is the product of p^e(p) then sigma_n(n) = Product_p ((p^(n*(e(p)+1)))-1)/(p^n-1).
sigma_n(n) is odd if and only if n is a square or twice a square. (End)
Conjecture: sigma_m(n) = sigma(n^m * rad(n)^(m-1))/sigma(rad(n)^(m-1)) for n > 0 and m > 0, where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 24 2017
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EXAMPLE
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The divisors of 6 are 1, 2, 3 and 6, so a(6) = 1^6 + 2^6 + 3^6 + 6^6 = 47450.
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MAPLE
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numtheory[sigma][n](n) ;
end proc:
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MATHEMATICA
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PROG
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(Python)
from sympy import divisor_sigma
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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