login
A100118
Numbers whose sum of prime factors is prime (counted with multiplicity).
33
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
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
KEYWORD
nonn
AUTHOR
Carlos Alves, Dec 26 2004
STATUS
approved