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%I #12 May 14 2018 07:53:26
%S 0,1,3,4,7,10,14,15,17,21,26,29,35,40,45,46,53,56,64,68,74,80,89,92,
%T 95,102,104,109,119,125,136,137,144,152,159,162,174,183,191,195,208,
%U 215,229,235,240,250,265,268,272,276,285,292,308,311,319,324,334,345,362,368,386,398,404,405,414
%N a(n) = Sum_{k=1..n} k*floor(n/prime(k)).
%C Partial sums of A066328.
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F G.f.: (1/(1 - x))*Sum_{k>=1} k*x^prime(k)/(1 - x^prime(k)).
%F a(p^k) = a(p^k-1) + pi(p), where p is a prime and pi() = A000720.
%F a(n) = A056239(A048803(n)).
%p seq(add(k*floor(n/ithprime(k)),k=1..n),n=1..65); # _Paolo P. Lava_, May 14 2018
%t Table[Sum[k Floor[n/Prime[k]], {k, n}], {n, 65}]
%t nmax = 65; Rest[CoefficientList[Series[1/(1 - x) Sum[k x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]
%t a[n_] := Plus @@ (PrimePi[#[[1]]] & /@ FactorInteger[n]); a[1] = 0; Accumulate[Table[a[n], {n, 65}]]
%Y Cf. A000040, A000720, A008472, A013939, A024916, A024924, A048803, A056239, A066328, A081401, A304038.
%K nonn
%O 1,3
%A _Ilya Gutkovskiy_, May 05 2018
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