

A046363


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


22



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
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OFFSET

1,1


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, p2p1=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


LINKS



FORMULA



EXAMPLE

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


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 *)


PROG

(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
(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


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



