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A332517
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a(n) = Sum_{k=1..n} gcd(n,k)^n.
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16
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1, 5, 29, 274, 3129, 47515, 823549, 16843268, 387459861, 10009769725, 285311670621, 8918311856102, 302875106592265, 11112685048729175, 437893951473411261, 18447025557276459016, 827240261886336764193, 39346558373052524325225, 1978419655660313589123997
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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a(n) = Sum_{d|n} phi(n/d) * d^n.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_(n-1)(d).
a(n) ~ n^n.
a(n) = Sum_{k=1..n} (n/gcd(n,k))^n*phi(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*gcd(n,k)*sigma_(n-1)(gcd(n,k))/phi(n/gcd(n,k)). (End)
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MAPLE
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f:= n -> add(igcd(n, k)^n, k=1..n):
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MATHEMATICA
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Table[Sum[GCD[n, k]^n, {k, 1, n}], {n, 1, 19}]
Table[Sum[EulerPhi[n/d] d^n, {d, Divisors[n]}], {n, 1, 19}]
Table[Sum[MoebiusMu[n/d] d DivisorSigma[n - 1, d], {d, Divisors[n]}], {n, 1, 19}]
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PROG
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(PARI) a(n) = sum(k=1, n, gcd(n, k)^n); \\ Michel Marcus, Feb 14 2020
(Magma) [&+[Gcd(n, k)^n:k in [1..n]]: n in [1..20]]; // Marius A. Burtea, Feb 15 2020
(Python)
from sympy import totient, divisors
return sum(totient(d)*(n//d)**n 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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