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A373000
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a(n) = Sum_{k=1..n} sigma( (n/gcd(k,n))^4 ).
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3
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1, 32, 243, 1054, 3125, 7776, 16807, 33818, 59289, 100000, 161051, 256122, 371293, 537824, 759375, 1082386, 1419857, 1897248, 2476099, 3293750, 4084101, 5153632, 6436343, 8217774, 9768745, 11881376, 14408187, 17714578, 20511149, 24300000, 28629151, 34636802, 39135393
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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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If k is squarefree (cf. A005117) then a(k) = k^5.
a(n) = Sum_{d|n} phi(d) * sigma(d^4).
Multiplicative with a(p^e) = (p^(5*e+5)-1)/(p^5-1) - (p^e-1)/(p-1).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = (zeta(5) * zeta(6) / zeta(2)) * Product_{p prime} (1 + 1/p^2 + 1/p^4 - 1/p^5) = 1.00815225456201682259... . (End)
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MATHEMATICA
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a[n_] := DivisorSum[n, EulerPhi[#] * DivisorSigma[1, #^4] &]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, eulerphi(d)*sigma(d^4));
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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