A282774 - OEIS

%I #25 Oct 11 2022 00:53:27

%S 1,8,50,128,228,9976,32768,41890,47668,53064,501888,564736,1207944,

%T 12026888,14697568,29720448,2147483648,2256502784,21471264576,

%U 35929849856

%N Nonprime numbers k such that sigma(k) - Sum_{j=1..m}{sigma(k) mod d_j} | k, where d_j is one of the m divisors of k.

%C For 1, 228, 501888, 1207944, 29720448, etc., being their ratio equal to 1, we have that Sum_{j=1..m}{sigma(k) mod d_j} is the sum of their aliquot parts.

%C The ratios for the listed terms are 1, 2, 2, 16, 1, 8, 2048, 2, 2, 22, 1, 512, 1, 25976, 32, 1, 67108864, 32768, ...

%C a(21) > 6 * 10^10. - _Lucas A. Brown_, Mar 10 2021

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A282774%2B5.py">A282774+5.py</a>

%e sigma(50) = 93; divisors of 50 are 1, 2, 5, 10, 25, 50 and

%e 93 mod 1 + 93 mod 2 + 93 mod 4 + 93 mod 5 + 93 mod 10 + 93 mod 25 + 93 mod 50 = 0 + 1 + 3 + 3 + 18 + 43 = 68 and 50 / (93-68) = 2.

%p with(numtheory): P:=proc(q) local a,b,c,k,n;

%p for n from 1 to q do if not isprime(n) then a:=sigma(n); b:=sort([op(divisors(n))]);

%p c:=add(a mod b[k],k=1..nops(b)); if type(n/(a-c),integer) then print(n); fi; fi; od; end: P(10^9);

%o (PARI) isok(k) = !isprime(k) && !(k % (sigma(k) - sumdiv(k, d, sigma(k) % d))); \\ _Michel Marcus_, Mar 10 2021

%Y Cf. A000203, A282775.

%K nonn,more

%O 1,2

%A _Paolo P. Lava_, Feb 22 2017

%E a(14)-a(18) from _Giovanni Resta_, Feb 23 2017

%E Name clarified and a(19)-a(20) from _Lucas A. Brown_, Mar 10 2021