A046346 Composite numbers that are divisible by the sum of their prime factors (counted with multiplicity).

4, 16, 27, 30, 60, 70, 72, 84, 105, 150, 180, 220, 231, 240, 256, 286, 288, 308, 378, 440, 450, 476, 528, 540, 560, 576, 588, 594, 624, 627, 646, 648, 650, 728, 800, 805, 840, 884, 897, 900, 945, 960, 1008, 1040, 1056, 1080, 1100, 1122, 1134, 1160, 1170, 1248

Offset: 1

Author

Patrick De Geest, Jun 15 1998

Keywords

• nonn

Comments

If m is in the sequence and d|m, then m^d is also a term. Note that this sequence contains all infinite subsequences of the form p^(p^k) for k>0, where p is a prime. - Amiram Eldar and Thomas Ordowski, Feb 06 2019

If one selects some composite k, k >= 8, and decomposes (k - sopfr(k)) into an additive partition having only prime parts, then those parts, when taken as a product with k, yield an element of this sequence. - Christopher Hohl, Jul 30 2019

Examples

			a(38) = 884 = 2 * 2 * 13 * 17 -> 2 + 2 + 13 + 17 = 34 so 884 / 34 = 26.
		

Links

• François Huppé, Table of n, a(n) for n = 1..50000 (terms 1..1000 from T. D. Noe)
• K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. See "special numbers" on page 287.

Crossrefs

Cf. A036844, A046347, A046348, A001414.

Contains A071142.

Programs

MATLAB

m=1;for u=2:1200 if and(isprime(u)==0,mod(u,sum(factor(u)))==0); sol(m)=u; m=m+1; end; end;sol % Marius A. Burtea, Jul 31 2019

Magma

[k:k in [2..1200]| not IsPrime(k) and  k mod (&+[m[1]*m[2]: m in Factorization(k)]) eq 0]; // Marius A. Burtea, Jul 31 2019

Maple

isA046346 := proc(n)
    if isprime(n) then
        false;
    elif modp(n,A001414(n)) = 0 then
        true;
    else
        false;
    end if;
end proc:
for n from 2 to 1000 do
    if isA046346(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016

Mathematica

Select[Range[2,1170],!PrimeQ[#]&&IntegerQ[#/Total[Times@@@FactorInteger[#]]]&] (* Jayanta Basu, Jun 02 2013 *)

PARI

sopfr(n) = {my(f=factor(n)); sum(k=1, #f~, f[k,1]*f[k,2]);}
lista(nn) = forcomposite(n=2, nn, if (! (n % sopfr(n)), print1(n, ", "));); \\ Michel Marcus, Jan 06 2016

Python

from sympy import factorint
def ok(n):
  f = factorint(n)
  return sum(f[p] for p in f) > 1 and n % sum(p*f[p] for p in f) == 0
print(list(filter(ok, range(1250)))) # Michael S. Branicky, Apr 16 2021

Extensions

Description corrected by Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 09 2002