A213741 Numbers n such that the sum of the first n primes is divisible by exactly 3 prime powers (not including 1).

5, 13, 20, 23, 24, 35, 39, 41, 42, 43, 47, 50, 56, 61, 62, 63, 67, 68, 69, 70, 73, 76, 78, 81, 86, 90, 98, 112, 123, 126, 128, 134, 143, 145, 147, 160, 165, 166, 172, 176, 180, 182, 186, 189, 191, 193, 196, 197, 200, 215, 220, 222, 223, 225, 227, 229, 238

Offset: 1

Author

Jonathan Vos Post, Jun 19 2012

Keywords

• nonn
• easy

Comments

This is to "triprimes" or "3-almost primes" A014612 as A213650 is to semiprimes A001358.

Examples

			a(1) = 5 because the sum of first 5 primes is 28 = 2^2 * 7 which has exactly three prime power factors (not including 1).
a(2) = 13 because the sum of first 13 primes is 238  =  2 * 7 * 17  which has exactly three prime power factors (not including 1).
a(3) = 20 because the sum of first 20 primes is 639 = 3^2 * 71.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007504, A014612, A213650.

Programs

Mathematica

ps = 0; t = {}; Do[ps = ps + Prime[n]; If[Total[Transpose[FactorInteger[ps]][[2]]] == 3, AppendTo[t, n]], {n, 300}]; t (* T. D. Noe, Jun 27 2012 *)

PARI

list(lim)=my(v=List(),k,s); forprime(p=2,prime(lim\1), k++; if(bigomega(s+=p)==3, listput(v,k))); Vec(v) \\ Charles R Greathouse IV, Feb 05 2017

Formula

{n such that A007504(n) is included in A014612.}