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

6, 10, 12, 22, 28, 34, 40, 45, 48, 52, 54, 56, 58, 63, 75, 76, 80, 82, 88, 90, 96, 99, 104, 108, 117, 118, 136, 142, 147, 148, 153, 165, 172, 175, 176, 184, 198, 202, 207, 210, 214, 224, 245, 248, 250, 252, 268, 273, 274, 279, 294, 296, 298, 300, 316, 320, 325

Offset: 1

Author

Patrick De Geest, Jun 15 1998

Keywords

• nonn

Comments

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

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

The ternary Goldbach Conjecture implies that this sequence contains infinitely many terms of A014612 (triprimes). - Elijah Beregovsky, Dec 17 2019

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

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

Examples

			214 = 2 * 107 -> Sum of factors is 109 -> 109 is prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A046364, A046365, A100118, A000040, A002808, A066038, A001414.

Programs

Magma

f:=func; [k:k in [2..350]| not IsPrime(k) and IsPrime(f(k))]; // Marius A. Burtea, Dec 17 2019

Maple

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

Mathematica

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 *)
Select[Range[325], !PrimeQ[#] && PrimeQ[Total[Times@@@FactorInteger[#]]]&] (* Jayanta Basu, May 29 2013 *)

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

Formula

A100118 INTERSECT A002808. - R. J. Mathar, Sep 09 2015

Extensions

Edited by R. J. Mathar, Nov 02 2009