login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Odd squarefree composite numbers k, divisible by the sum of their prime factors, sopfr (A001414).
2

%I #25 Sep 08 2022 08:46:21

%S 105,231,627,805,897,1581,2967,3055,4543,5487,6461,6745,7881,9717,

%T 10707,14231,15015,16377,21091,26331,29607,33495,33901,33915,35905,

%U 37411,38843,40587,42211,45885,49335,50505,51051,53295,55581,60297

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

%C 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 Some Carmichael numbers appear in this sequence, the first of which is 3240392401.

%C From _Robert Israel_, Jul 05 2019: (Start)

%C 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 Robert Israel, <a href="/A308643/b308643.txt">Table of n, a(n) for n = 1..10000</a>

%e 105=3*5*7; sum of prime factors = 15 and 105 = 7*15, so 105 is a term.

%p with(NumberTheory);

%p N := 500;

%p for n from 2 to N do

%p S := PrimeFactors(n);

%p X := add(S);

%p if IsSquareFree(n) and not mod(n, 2) = 0 and not isprime(n) and mod(n, X) = 0 then print(n);

%p end if:

%p end do:

%t 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 (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 Intersection of A005117 and A046347.

%Y Cf. A001414, A046346, A002997, A001221, A063908, A156300, A153135.

%K nonn

%O 1,1

%A _David James Sycamore_, Jun 13 2019