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

1,2

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

Links

Table of n, a(n) for n=1..28.

Lucas A. Brown, A282774+5.py

Example

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.

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);

Prog

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

Crossrefs

Cf. A000203, A007691, A282774.

Sequence in context: A259307 A183019 A183016 * A325808 A192853 A027598

Adjacent sequences: A282772 A282773 A282774 * A282776 A282777 A282778

Keyword

nonn

Author

Paolo P. Lava, Feb 22 2017

Extensions

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

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

Status

approved