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

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from indices in prime factorization

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.

Example

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

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}]

Crossrefs

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

Sequence in context: A200113 A065489 A346404 * A347968 A197258 A211783

Adjacent sequences: A332382 A332383 A332384 * A332386 A332387 A332388

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 10 2020

Status

approved