A332385 Sum of squares of indices of distinct prime factors of n.

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

Offset: 1

Author

Ilya Gutkovskiy, Feb 10 2020

Keywords

• nonn

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

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

Programs

Maple

a:= n-> add(numtheory[pi](i[1])^2, i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 10 2020

Mathematica

nmax = 75; CoefficientList[Series[Sum[k^2 x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Plus @@ (PrimePi[#[[1]]]^2 & /@ FactorInteger[n]); Table[a[n], {n, 1, 75}]

Formula

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

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