A046364 -id:A046364 - cp's OEIS frontend

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

Patrick De Geest, Jun 15 1998

nonn

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

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

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

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

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

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

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 *)

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

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

Edited by R. J. Mathar, Nov 02 2009

A046365 Composite palindromes whose sum of prime factors is prime (counted with multiplicity).

Original entry on oeis.org

6, 22, 88, 99, 202, 252, 333, 414, 424, 454, 464, 595, 686, 747, 777, 808, 838, 848, 858, 909, 1001, 1551, 1771, 2442, 3553, 4114, 5335, 5775, 6336, 6996, 8008, 8228, 9009, 9559, 9669, 9889, 12121, 14241, 16261, 16761, 17171, 18081, 18381, 20102, 20602, 21012

Offset: 1

Patrick De Geest, Jun 15 1998

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A046363, A046364.

Select[Range[20125], !PrimeQ[#] && Reverse[x=IntegerDigits[#]] == x && PrimeQ[Total[Times@@@FactorInteger[#]]]&] (* Jayanta Basu, May 29 2013 *)

from itertools import product
from sympy import factorint, isprime
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(pal):
    f = factorint(pal); return len(f)>1 and isprime(sum(p*f[p] for p in f))
print(list(filter(ok, (p for d in range(1, 6) for p in pals(d) if ok(p))))) # Michael S. Branicky, Jun 22 2021

A046363 INTERSECT A002113. - R. J. Mathar, Sep 09 2015

a(45) and beyond from Michael S. Branicky, Jun 22 2021

cp's OEIS Frontend

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

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