A100118 Numbers whose sum of prime factors is prime (counted with multiplicity).

2, 3, 5, 6, 7, 10, 11, 12, 13, 17, 19, 22, 23, 28, 29, 31, 34, 37, 40, 41, 43, 45, 47, 48, 52, 53, 54, 56, 58, 59, 61, 63, 67, 71, 73, 75, 76, 79, 80, 82, 83, 88, 89, 90, 96, 97, 99, 101, 103, 104, 107, 108, 109, 113, 117, 118, 127, 131, 136, 137, 139, 142, 147, 148, 149

Offset: 1

Author

Carlos Alves, Dec 26 2004

Keywords

• nonn

Comments

Numbers n such that integer log of n is a prime number.

As in A001414, denote sopfr(n) the integer log of n. Since sopfr(p)=p, the sequence includes all prime numbers.

See A046363 for the analog excluding prime numbers. - Hieronymus Fischer, Oct 20 2007

These numbers may be arranged in a family of posets of triangles of multiarrows (see link and example). - Gus Wiseman, Sep 14 2016

Examples

			40 = 2^3*5 and 2*3 + 5 = 11 is a prime number.
These numbers correspond to multiarrows in the multiorder of partitions of prime numbers into prime parts. For example: 2:2<=(2), 3:3<=(3), 6:5<=(2,3), 5:5<=(5), 12:7<=(2,2,3), 10:7<=(2,5), 7:7<=(7), 48:11<=(2,2,2,2,3), 52:11<=(2,3,3,3), 40:11<=(2,2,2,5), 45:11<=(3,3,5), 28:11<=(2,2,7), 11:11<=(11). - _Gus Wiseman_, Sep 14 2016
		

Links

• Jayanta Basu, Table of n, a(n) for n = 1..10000
• Gus Wiseman, Lattice form posets indexed by A100118

Crossrefs

Cf. A001414, A046363, A056768, A276687.

Programs

Maple

for n from 1 to 200 do
    if isprime(A001414(n)) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Sep 09 2015

Mathematica

L = {}; Do[ww = Transpose[FactorInteger[k]];w = ww[[1]].ww[[2]]; If[PrimeQ[w], AppendTo[L, k]], {k, 2, 500}];L
Select[Range[150], PrimeQ[Total[Times @@@ FactorInteger[#]]] &] (* Jayanta Basu, Aug 11 2013 *)

PARI

is(n)=my(f=factor(n)); isprime(sum(i=1,#f~,f[i,1]*f[i,2])) \\ Charles R Greathouse IV, Sep 21 2013