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

%I A100118 #33 Sep 20 2020 20:58:05
%S A100118 2,3,5,6,7,10,11,12,13,17,19,22,23,28,29,31,34,37,40,41,43,45,47,48,
%T A100118 52,53,54,56,58,59,61,63,67,71,73,75,76,79,80,82,83,88,89,90,96,97,99,
%U A100118 101,103,104,107,108,109,113,117,118,127,131,136,137,139,142,147,148,149
%N A100118 Numbers whose sum of prime factors is prime (counted with multiplicity).
%C A100118 Numbers n such that integer log of n is a prime number.
%C A100118 As in A001414, denote sopfr(n) the integer log of n. Since sopfr(p)=p, the sequence includes all prime numbers.
%C A100118 See A046363 for the analog excluding prime numbers. - _Hieronymus Fischer_, Oct 20 2007
%C A100118 These numbers may be arranged in a family of posets of triangles of multiarrows (see link and example). - _Gus Wiseman_, Sep 14 2016
%H A100118 Jayanta Basu, <a href="/A100118/b100118.txt">Table of n, a(n) for n = 1..10000</a>
%H A100118 Gus Wiseman, <a href="/A100118/a100118.png">Lattice form posets indexed by A100118</a>
%e A100118 40 = 2^3*5 and 2*3 + 5 = 11 is a prime number.
%e A100118 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
%p A100118 for n from 1 to 200 do
%p A100118     if isprime(A001414(n)) then
%p A100118         printf("%d,",n);
%p A100118     end if;
%p A100118 end do: # _R. J. Mathar_, Sep 09 2015
%t A100118 L = {}; Do[ww = Transpose[FactorInteger[k]];w = ww[[1]].ww[[2]]; If[PrimeQ[w], AppendTo[L, k]], {k, 2, 500}];L
%t A100118 Select[Range[150], PrimeQ[Total[Times @@@ FactorInteger[#]]] &] (* _Jayanta Basu_, Aug 11 2013 *)
%o A100118 (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
%Y A100118 Cf. A001414, A046363, A056768, A276687.
%K A100118 nonn
%O A100118 1,1
%A A100118 _Carlos Alves_, Dec 26 2004