A282775 Nonprime numbers k such that k | (sigma(k) - Sum_{j=1..m}{sigma(k) mod d_j}), where d_j is one of the m divisors of k.

1, 6, 28, 120, 228, 496, 672, 8128, 30240, 32760, 125640, 501888, 523776, 1207944, 2178540, 23569920, 29720448, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160

Offset: 1

Author

Paolo P. Lava, Feb 22 2017

Keywords

• nonn

Comments

The multiply-perfect numbers are a subset.

For 1, 228, 501888, 1207944, 29720448, etc., their ratio being 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, 3, 1, 2, 3, 2, 4, 4, 2, 1, 3, 1, 4, 4, 1, 2, 4, 4, 3, 4, 3, 2, ...

a(29) > 6 * 10^10. - Lucas A. Brown, Mar 10 2021

Examples

			sigma(228) = 560; divisors of 288 are 1, 2, 3, 4, 6, 12, 19, 38, 57, 76, 114, 228 and 560 mod 1 + 560 mod 2 + 560 mod 3 + 560 mod 4 + ... + 560 mod 57 + 560 mod 76 + 560 mod 144 + 560 mod 228 = 0 + 0 + 2 + 0 + 2 + 8 + 9 + 28 + 47 + 28 + 104 + 104 = 332 and (560 - 332) / 228 = 1.
		

Links

• Lucas A. Brown, A282774+5.py

Crossrefs

Cf. A000203, A007691, A282774.

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((a-c)/n,integer) then print(n); fi; fi; od; end: P(10^9);

PARI

isok(k) = if (!isprime(k), my(sk = sigma(k)); (sk - sumdiv(k, d, sk % d)) % k == 0;); \\ Michel Marcus, Jun 17 2017

Extensions

a(16)-a(24) from Giovanni Resta, Feb 23 2017

a(25)-a(28) from Lucas A. Brown, Mar 10 2021