A372227 - OEIS

%I #37 May 20 2024 02:30:12

%S 1,8,27,70,125,216,343,578,753,1000,1331,1890,2197,2744,3375,4666,

%T 4913,6024,6859,8750,9261,10648,12167,15606,15745,17576,20427,24010,

%U 24389,27000,29791,37418,35937,39304,42875,52710,50653,54872,59319,72250,68921,74088

%N a(n) = Sum_{k=1..n} sigma( (n/gcd(k,n))^2 ).

%H Seiichi Manyama, <a href="/A372227/b372227.txt">Table of n, a(n) for n = 1..10000</a>

%F If k is squarefree (cf. A005117) then a(k) = k^3.

%F a(n) = Sum_{d|n} phi(d) * sigma(d^2).

%F From _Amiram Eldar_, May 20 2024: (Start)

%F Multiplicative with a(p^e) = (p^(3*e+3)-1)/(p^3-1) - (p^e-1)/(p-1).

%F 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)

%t a[n_] := DivisorSum[n, EulerPhi[#] * DivisorSigma[1, #^2] &]; Array[a, 100] (* _Amiram Eldar_, May 20 2024 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(d)*sigma(d^2));

%Y Cf. A062952, A372998, A373000.

%Y Cf. A062380, A372996.

%Y Cf. A005117.

%Y Cf. A002117, A182448.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 19 2024