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

1,1

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

Links

Jayanta Basu, Table of n, a(n) for n = 1..10000

Gus Wiseman, Lattice form posets indexed by A100118

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A001414, A046363, A056768, A276687.

Sequence in context: A114518 A066940 A305318 * A295389 A028781 A136149

Adjacent sequences: A100115 A100116 A100117 * A100119 A100120 A100121

Keyword

nonn

Author

Carlos Alves, Dec 26 2004

Status

approved