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A372998
a(n) = Sum_{k=1..n} sigma( (n/gcd(k,n))^3 ).
3
1, 16, 81, 270, 625, 1296, 2401, 4362, 6639, 10000, 14641, 21870, 28561, 38416, 50625, 69890, 83521, 106224, 130321, 168750, 194481, 234256, 279841, 353322, 391245, 456976, 538071, 648270, 707281, 810000, 923521, 1118450, 1185921, 1336336, 1500625, 1792530
OFFSET
1,2
LINKS
FORMULA
If k is squarefree (cf. A005117) then a(k) = k^4.
a(n) = Sum_{d|n} phi(d) * sigma(d^3).
From Amiram Eldar, May 20 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+4)-1)/(p^4-1) - (p^e-1)/(p-1).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(4) * zeta(5) * Product_{p prime} (1 - 1/p^4 - 1/p^5 + 1/p^6) = 1.01649108704844291655... . (End)
MATHEMATICA
a[n_] := DivisorSum[n, EulerPhi[#] * DivisorSigma[1, #^3] &]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)
PROG
(PARI) a(n) = sumdiv(n, d, eulerphi(d)*sigma(d^3));
CROSSREFS
Cf. A005117.
Sequence in context: A113317 A187457 A056118 * A351602 A134606 A343324
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 19 2024
STATUS
approved