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A046363 Composite numbers whose sum of prime factors (with multiplicity) is prime.

%I A046363 #68 Feb 26 2024 10:28:53
%S A046363 6,10,12,22,28,34,40,45,48,52,54,56,58,63,75,76,80,82,88,90,96,99,104,
%T A046363 108,117,118,136,142,147,148,153,165,172,175,176,184,198,202,207,210,
%U A046363 214,224,245,248,250,252,268,273,274,279,294,296,298,300,316,320,325
%N A046363 Composite numbers whose sum of prime factors (with multiplicity) is prime.
%C A046363 If prime numbers were included the sequence would be 2, 3, 5, 6, 7, 10, 11, 12, 13, 17, 19, 22, 23, 28, 29, ... which is A100118. - _Hieronymus Fischer_, Oct 20 2007
%C A046363 Conjecture: a(n) can be approximated with the formula c*n^k, where c is approximately 0.46 and k is approximately 1.05. - _Elijah Beregovsky_, May 01 2019
%C A046363 The ternary Goldbach Conjecture implies that this sequence contains infinitely many terms of A014612 (triprimes). - _Elijah Beregovsky_, Dec 17 2019
%C A046363 A proof that this sequence is infinite: There are infinitely many odd primes, let p2 > p1 > 2 be two odd primes, p2-p1=2*k  then (2^k)*p1 is a term because 2*k+p1=p2 is prime. For example: 5+6=11, 6=2*3, 2^3*5=40 is a term. - _Metin Sariyar_, Dec 17 2019
%C A046363 Regarding the 2019 conjecture, with k the same, the correct value of "c" is greater than 5, based on data to n = 10^7. - _Bill McEachen_, Feb 17 2024
%H A046363 Robert Israel, <a href="/A046363/b046363.txt">Table of n, a(n) for n = 1..10000</a>
%F A046363 A100118 INTERSECT A002808. - _R. J. Mathar_, Sep 09 2015
%e A046363 214 = 2 * 107 -> Sum of factors is 109 -> 109 is prime.
%p A046363 ifac := proc (n) local L, x: L := ifactors(n)[2]: map(proc (x) options operator, arrow: seq(x[1], j = 1 .. x[2]) end proc, L) end proc: a := proc (n) if isprime(n) = false and isprime(add(t, t = ifac(n))) = true then n else end if end proc: seq(a(n), n = 1 .. 350); # with help from _W. Edwin Clark_ - _Emeric Deutsch_, Jan 21 2009
%t A046363 PrimeFactorsAdded[n_] := Plus @@ Flatten[Table[ #[[1]]*#[[2]], {1}] & /@ FactorInteger[n]]; GenerateA046363[n_] := Select[Range[n], PrimeQ[PrimeFactorsAdded[ # ]] && PrimeQ[ # ] == False &]; (* GenerateA046363[100] would give all elements of this sequence below 100. - Ryan Witko (witko(AT)nyu.edu), Mar 08 2004 *)
%t A046363 Select[Range[325], !PrimeQ[#] && PrimeQ[Total[Times@@@FactorInteger[#]]]&] (* _Jayanta Basu_, May 29 2013 *)
%o A046363 (PARI) is(n)=if(isprime(n),return(0)); my(f=factor(n)); isprime(sum(i=1,#f~,f[i,1]*f[i,2])) \\ _Charles R Greathouse IV_, Sep 21 2013
%o A046363 (Magma) f:=func<n|&+[j[1]*j[2]: j in Factorization(n)]>; [k:k in [2..350]| not IsPrime(k) and IsPrime(f(k))]; // _Marius A. Burtea_, Dec 17 2019
%Y A046363 Cf. A046364, A046365, A100118, A000040, A002808, A066038, A001414.
%K A046363 nonn
%O A046363 1,1
%A A046363 _Patrick De Geest_, Jun 15 1998
%E A046363 Edited by _R. J. Mathar_, Nov 02 2009