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A321294
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a(n) = Sum_{d|n} mu(n/d)*d*sigma_n(d).
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3
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1, 9, 83, 1058, 15629, 282381, 5764807, 134480900, 3486902505, 100048836321, 3138428376731, 107006403495850, 3937376385699301, 155572843119518781, 6568408661060858767, 295150157013526773768, 14063084452067724991025, 708236697425777157039381
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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) = [x^n] Sum_{i>=1} Sum_{j>=1} mu(i)*j^(n+1)*x^(i*j)/(1 - x^(i*j))^2.
a(n) = Sum_{d|n} phi(n/d)*d^(n+1).
a(n) = Sum_{k=1..n} gcd(n,k)^(n+1).
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MATHEMATICA
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Table[Sum[MoebiusMu[n/d] d DivisorSigma[n, d], {d, Divisors[n]}], {n, 18}]
Table[Sum[EulerPhi[n/d] d^(n + 1), {d, Divisors[n]}], {n, 18}]
Table[Sum[GCD[n, k]^(n + 1), {k, n}], {n, 18}]
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PROG
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(PARI) a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, n)); \\ Michel Marcus, Nov 03 2018
(Python)
from sympy import totient, divisors
return sum(totient(d)*(n//d)**(n+1) for d in divisors(n, generator=True)) # Chai Wah Wu, Feb 15 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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