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A372620
Expansion of Sum_{k>=1} k * prime(k) * x^prime(k) / (1 - x^prime(k)).
0
0, 2, 6, 2, 15, 8, 28, 2, 6, 17, 55, 8, 78, 30, 21, 2, 119, 8, 152, 17, 34, 57, 207, 8, 15, 80, 6, 30, 290, 23, 341, 2, 61, 121, 43, 8, 444, 154, 84, 17, 533, 36, 602, 57, 21, 209, 705, 8, 28, 17, 125, 80, 848, 8, 70, 30, 158, 292, 1003, 23, 1098, 343, 34, 2, 93
OFFSET
1,2
FORMULA
L.g.f.: -log( Product_{k>=1} (1 - x^prime(k))^k ).
If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j) * p_j), where pi = A000720.
EXAMPLE
a(60) = a(2^2 * 3 * 5) = a(prime(1)^2 * prime(2) * prime(3)) = 1 * 2 + 2 * 3 + 3 * 5 = 23.
MATHEMATICA
nmax = 65; CoefficientList[Series[Sum[k Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Plus @@ (PrimePi[#[[1]]] #[[1]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 65}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 07 2024
STATUS
approved