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A046346 Composite numbers that are divisible by the sum of their prime factors (counted with multiplicity). 23
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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.
EXAMPLE
a(38) = 884 = 2 * 2 * 13 * 17 -> 2 + 2 + 13 + 17 = 34 so 884 / 34 = 26.
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 *)
PROG
(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
(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
(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
CROSSREFS
Contains A071142.
Sequence in context: A105078 A050707 A134330 * A340852 A328415 A097764
KEYWORD
nonn
AUTHOR
Patrick De Geest, Jun 15 1998
EXTENSIONS
Description corrected by Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 09 2002
STATUS
approved

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Last modified March 29 09:32 EDT 2024. Contains 371268 sequences. (Running on oeis4.)