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A308643
Odd squarefree composite numbers k, divisible by the sum of their prime factors, sopfr (A001414).
2
105, 231, 627, 805, 897, 1581, 2967, 3055, 4543, 5487, 6461, 6745, 7881, 9717, 10707, 14231, 15015, 16377, 21091, 26331, 29607, 33495, 33901, 33915, 35905, 37411, 38843, 40587, 42211, 45885, 49335, 50505, 51051, 53295, 55581, 60297
OFFSET
1,1
COMMENTS
Every term has an odd number of prime divisors (A001221(k) is odd), since if not, sopfr(k) would be even, and so not divide k, which is odd.
Some Carmichael numbers appear in this sequence, the first of which is 3240392401.
From Robert Israel, Jul 05 2019: (Start)
Includes p*q*r if p and q are distinct odd primes and r=(p-1)*(q-1)-1 is prime. Dickson's conjecture implies that there are infinitely many such terms for each odd prime p. Thus for p=3, q is in A063908 (except 3), for p=5, q is in A156300 (except 2), and for p=7, q is in A153135 (except 2). (End)
LINKS
EXAMPLE
105=3*5*7; sum of prime factors = 15 and 105 = 7*15, so 105 is a term.
MAPLE
with(NumberTheory);
N := 500;
for n from 2 to N do
S := PrimeFactors(n);
X := add(S);
if IsSquareFree(n) and not mod(n, 2) = 0 and not isprime(n) and mod(n, X) = 0 then print(n);
end if:
end do:
MATHEMATICA
aQ[n_] := Module[{f = FactorInteger[n]}, p=f[[;; , 1]]; e=f[[;; , 2]]; Length[e] > 1 && Max[e]==1 && Divisible[n, Plus@@(p^e)]]; Select[Range[1, 61000, 2], aQ] (* Amiram Eldar, Jul 04 2019 *)
PROG
(Magma) [k:k in [2*d+1: d in [1..35000]]|IsSquarefree(k) and not IsPrime(k) and k mod &+PrimeDivisors(k) eq 0]; // Marius A. Burtea, Jun 19 2019
CROSSREFS
Intersection of A005117 and A046347.
Sequence in context: A176878 A088595 A375168 * A229094 A307108 A262723
KEYWORD
nonn
AUTHOR
STATUS
approved