|
| |
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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, 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
|
|
|
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
|
| |
|
|
