A372226 - OEIS

%I #29 May 20 2024 02:29:46

%S 1,6,21,48,105,126,301,388,567,630,1221,1008,2041,1806,2205,3116,4641,

%T 3402,6517,5040,6321,7326,11661,8148,13125,12246,15327,14448,23577,

%U 13230,28861,24956,25641,27846,31605,27216,49321,39102,42861,40740,67281,37926

%N a(n) = Sum_{k=1..n} sigma_2( n/gcd(k,n) ).

%H Amiram Eldar, <a href="/A372226/b372226.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} phi(d) * sigma_2(d).

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

%F Multiplicative with a(p^e) = (p^(3*e+5) - p^(3*e+4) - p^(e+3) + p^e + p^4 - p^2) / ((p^2 - 1) * (p^3 - 1)).

%F Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.749582840863254826301... . (End)

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

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

%Y Cf. A062949, A062952.

%Y Cf. A064987.

%Y Cf. A002117, A013662.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 19 2024