A304066 a(n) = Sum_{k=1..n} k*floor(n/prime(k)).

0, 1, 3, 4, 7, 10, 14, 15, 17, 21, 26, 29, 35, 40, 45, 46, 53, 56, 64, 68, 74, 80, 89, 92, 95, 102, 104, 109, 119, 125, 136, 137, 144, 152, 159, 162, 174, 183, 191, 195, 208, 215, 229, 235, 240, 250, 265, 268, 272, 276, 285, 292, 308, 311, 319, 324, 334, 345, 362, 368, 386, 398, 404, 405, 414

Offset: 1

Author

Ilya Gutkovskiy, May 05 2018

Keywords

• nonn

Comments

Partial sums of A066328.

Links

• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000040, A000720, A008472, A013939, A024916, A024924, A048803, A056239, A066328, A081401, A304038.

Programs

Maple

seq(add(k*floor(n/ithprime(k)),k=1..n),n=1..65); # Paolo P. Lava, May 14 2018

Mathematica

Table[Sum[k Floor[n/Prime[k]], {k, n}], {n, 65}]
nmax = 65; Rest[CoefficientList[Series[1/(1 - x) Sum[k x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Plus @@ (PrimePi[#[[1]]] & /@ FactorInteger[n]); a[1] = 0; Accumulate[Table[a[n], {n, 65}]]

Formula

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

a(p^k) = a(p^k-1) + pi(p), where p is a prime and pi() = A000720.

a(n) = A056239(A048803(n)).