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A322664 a(n) = n^2 * Sum_{p^k|n} Sum_{j=1..k} 1/p^(2*j), where p are primes dividing n with multiplicity k.

%I A322664 #17 Jul 25 2025 19:07:23
%S A322664 0,1,1,5,1,13,1,21,10,29,1,61,1,53,34,85,1,121,1,141,58,125,1,253,26,
%T A322664 173,91,261,1,361,1,341,130,293,74,565,1,365,178,589,1,673,1,621,331,
%U A322664 533,1,1021,50,729,298,861,1,1093,146,1093,370,845,1,1669,1,965
%N A322664 a(n) = n^2 * Sum_{p^k|n} Sum_{j=1..k} 1/p^(2*j), where p are primes dividing n with multiplicity k.
%C A322664 The generalized formula is f(n,m) = n^m * Sum_{p^k|n} Sum_{j=1..k} 1/p^(m*j), where f(n,0) = A001222(n) and f(n,1) = A095112(n).
%C A322664 From _Ridouane Oudra_, Jul 21 2025: (Start)
%C A322664 a(n) is the sum of (n/d)^2 over all prime powers d which divide n.
%C A322664 Using the previous generalized formula we have :
%C A322664 f(n,m) = Sum_{d|n, d is a prime power} (n/d)^m.
%C A322664 f(n,m) = Sum_{d|n} bigomega(d)*J_m(n/d), where J_m is the m-th Jordan totient function. (End)
%H A322664 Antti Karttunen, <a href="/A322664/b322664.txt">Table of n, a(n) for n = 1..20000</a>
%F A322664 Sum_{k=1..n} a(k) ~ A286229 * A000330(n).
%F A322664 a(n) = Sum_{d|n} bigomega(d)*J_2(n/d), where J_2 = A007434. - _Ridouane Oudra_, Jul 21 2025
%e A322664 The prime factorization of 24 is 2^3 * 3, so a(24) = 24^2 * (1/2^2 + 1/2^(2*2) + 1/2^(2*3) + 1/3^2) = 253.
%o A322664 (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, sum(j=1, f[k,2], n^2 / f[k,1]^(2*j)));
%Y A322664 Cf. A001222, A095112, A286229, A007434.
%K A322664 nonn
%O A322664 1,4
%A A322664 _Daniel Suteu_, Dec 22 2018