A372227 a(n) = Sum_{k=1..n} sigma( (n/gcd(k,n))^2 ).
1, 8, 27, 70, 125, 216, 343, 578, 753, 1000, 1331, 1890, 2197, 2744, 3375, 4666, 4913, 6024, 6859, 8750, 9261, 10648, 12167, 15606, 15745, 17576, 20427, 24010, 24389, 27000, 29791, 37418, 35937, 39304, 42875, 52710, 50653, 54872, 59319, 72250, 68921, 74088
Offset: 1
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
a[n_] := DivisorSum[n, EulerPhi[#] * DivisorSigma[1, #^2] &]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)
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PARI
a(n) = sumdiv(n, d, eulerphi(d)*sigma(d^2));
Formula
If k is squarefree (cf. A005117) then a(k) = k^3.
a(n) = Sum_{d|n} phi(d) * sigma(d^2).
From Amiram Eldar, May 20 2024: (Start)
Multiplicative with a(p^e) = (p^(3*e+3)-1)/(p^3-1) - (p^e-1)/(p-1).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = (Pi^2/15) * zeta(3) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.03291869994469216597... . (End)