A088821 a(n) is the sum of smallest prime factors of numbers from 1 to n.

0, 2, 5, 7, 12, 14, 21, 23, 26, 28, 39, 41, 54, 56, 59, 61, 78, 80, 99, 101, 104, 106, 129, 131, 136, 138, 141, 143, 172, 174, 205, 207, 210, 212, 217, 219, 256, 258, 261, 263, 304, 306, 349, 351, 354, 356, 403, 405, 412, 414, 417, 419, 472, 474, 479, 481, 484

Offset: 1

Author

Labos Elemer, Oct 22 2003

Keywords

• nonn

References

• M. Kalecki, On certain sums extended over primes or prime factors, Prace Mat, Vol. 8 (1963), pp. 121-127.
• J. Sandor, D. S. Mitrinovic, B. Crstici, Handbook of Number Theory I, Volume 1, Springer, 2005, Chapter IV, p. 121.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Project Euler, Problem 521 - Smallest prime factor

Crossrefs

Cf. A020639, A046669, A088822, A088823, A088824, A088825.

Programs

GAP

P:=List(List([2..60],n->Factors(n)),i->i[1]);;
a:=Concatenation([0],List([1..Length(P)],i->Sum([1..i],k->P[k]))); # Muniru A Asiru, Nov 29 2018

Mathematica

Prepend[Accumulate[Rest[Table[FactorInteger[i][[1,1]],{i,60}]]],0] (* Harvey P. Dale, Jan 09 2011 *)

PARI

a(n) = sum(k=2, n, factor(k)[1,1]); \\ Michel Marcus, May 15 2017

Formula

a(n) ~ n^2/(2 log n) [Kalecki]. - Thomas Ordowski, Nov 29 2018

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