A332385 - OEIS

%I #9 Jun 28 2022 16:38:41

%S 0,1,4,1,9,5,16,1,4,10,25,5,36,17,13,1,49,5,64,10,20,26,81,5,9,37,4,

%T 17,100,14,121,1,29,50,25,5,144,65,40,10,169,21,196,26,13,82,225,5,16,

%U 10,53,37,256,5,34,17,68,101,289,14,324,122,20,1,45,30,361,50,85,26,400,5,441,145,13

%N Sum of squares of indices of distinct prime factors of n.

%H Alois P. Heinz, <a href="/A332385/b332385.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F G.f.: Sum_{k>=1} k^2 * x^prime(k) / (1 - x^prime(k)).

%F If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^2), where pi = A000720.

%e a(21) = a(3 * 7) = a(prime(2) * prime(4)) = 2^2 + 4^2 = 20.

%p a:= n-> add(numtheory[pi](i[1])^2, i=ifactors(n)[2]):

%p seq(a(n), n=1..80);  # _Alois P. Heinz_, Feb 10 2020

%t nmax = 75; CoefficientList[Series[Sum[k^2 x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t a[n_] := Plus @@ (PrimePi[#[[1]]]^2 & /@ FactorInteger[n]); Table[a[n], {n, 1, 75}]

%Y Cf. A000720, A005063, A056239, A066328, A067666, A090885, A289506.

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, Feb 10 2020