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
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
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