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A308643 Odd squarefree composite numbers k, divisible by the sum of their prime factors, sopfr (A001414).

%I A308643 #25 Sep 08 2022 08:46:21
%S A308643 105,231,627,805,897,1581,2967,3055,4543,5487,6461,6745,7881,9717,
%T A308643 10707,14231,15015,16377,21091,26331,29607,33495,33901,33915,35905,
%U A308643 37411,38843,40587,42211,45885,49335,50505,51051,53295,55581,60297
%N A308643 Odd squarefree composite numbers k, divisible by the sum of their prime factors, sopfr (A001414).
%C A308643 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.
%C A308643 Some Carmichael numbers appear in this sequence, the first of which is 3240392401.
%C A308643 From _Robert Israel_, Jul 05 2019: (Start)
%C A308643 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)
%H A308643 Robert Israel, <a href="/A308643/b308643.txt">Table of n, a(n) for n = 1..10000</a>
%e A308643 105=3*5*7; sum of prime factors = 15 and 105 = 7*15, so 105 is a term.
%p A308643 with(NumberTheory);
%p A308643 N := 500;
%p A308643 for n from 2 to N do
%p A308643 S := PrimeFactors(n);
%p A308643 X := add(S);
%p A308643 if IsSquareFree(n) and not mod(n, 2) = 0 and not isprime(n) and mod(n, X) = 0 then print(n);
%p A308643 end if:
%p A308643 end do:
%t A308643 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 *)
%o A308643 (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
%Y A308643 Intersection of A005117 and A046347.
%Y A308643 Cf. A001414, A046346, A002997, A001221, A063908, A156300, A153135.
%K A308643 nonn
%O A308643 1,1
%A A308643 _David James Sycamore_, Jun 13 2019