A282774 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.

1, 8, 50, 128, 228, 9976, 32768, 41890, 47668, 53064, 501888, 564736, 1207944, 12026888, 14697568, 29720448, 2147483648, 2256502784, 21471264576, 35929849856

Offset: 1

Author

Paolo P. Lava, Feb 22 2017

Keywords

• nonn
• more

Comments

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.

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, ...

a(21) > 6 * 10^10. - Lucas A. Brown, Mar 10 2021

Examples

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

Links

• Lucas A. Brown, A282774+5.py

Crossrefs

Cf. A000203, A282775.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,k,n;
for n from 1 to q do if not isprime(n) then a:=sigma(n); b:=sort([op(divisors(n))]);
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);

PARI

isok(k) = !isprime(k) && !(k % (sigma(k) - sumdiv(k, d, sigma(k) % d))); \\ Michel Marcus, Mar 10 2021

Extensions

a(14)-a(18) from Giovanni Resta, Feb 23 2017

Name clarified and a(19)-a(20) from Lucas A. Brown, Mar 10 2021