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%I #43 Dec 13 2018 04:41:40
%S 0,2,5,7,12,14,21,23,26,28,39,41,54,56,59,61,78,80,99,101,104,106,129,
%T 131,136,138,141,143,172,174,205,207,210,212,217,219,256,258,261,263,
%U 304,306,349,351,354,356,403,405,412,414,417,419,472,474,479,481,484
%N a(n) is the sum of smallest prime factors of numbers from 1 to n.
%D M. Kalecki, On certain sums extended over primes or prime factors, Prace Mat, Vol. 8 (1963), pp. 121-127.
%D J. Sandor, D. S. Mitrinovic, B. Crstici, Handbook of Number Theory I, Volume 1, Springer, 2005, Chapter IV, p. 121.
%H Michael De Vlieger, <a href="/A088821/b088821.txt">Table of n, a(n) for n = 1..10000</a>
%H Project Euler, <a href="https://projecteuler.net/problem=521">Problem 521 - Smallest prime factor</a>
%F a(n) ~ n^2/(2 log n) [Kalecki]. - _Thomas Ordowski_, Nov 29 2018
%F a(n) = Sum_{prime p} n(p)*p, where n(p) is the number of integers in [1,n] with smallest prime factor spf(.) = A020639(.) = p, decreasing from n(2) = floor(n/2) to n(p) = 1 for p >= sqrt(n), possibly earlier, and n(p) = 0 for p > n. One has n(p) ~ D(p)*n where D(p) = (Product_{primes q < p} 1-1/q)/p = A038110/A038111 is the density of numbers having p as smallest prime factor. - _M. F. Hasler_, Dec 05 2018
%t Prepend[Accumulate[Rest[Table[FactorInteger[i][[1,1]],{i,60}]]],0] (* _Harvey P. Dale_, Jan 09 2011 *)
%o (PARI) a(n) = sum(k=2, n, factor(k)[1,1]); \\ _Michel Marcus_, May 15 2017
%o (GAP) P:=List(List([2..60],n->Factors(n)),i->i[1]);;
%o a:=Concatenation([0],List([1..Length(P)],i->Sum([1..i],k->P[k]))); # _Muniru A Asiru_, Nov 29 2018
%Y Cf. A020639, A046669, A088822, A088823, A088824, A088825.
%K nonn
%O 1,2
%A _Labos Elemer_, Oct 22 2003
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