cp's OEIS Frontend

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.

%I A282775 #35 Oct 11 2022 00:54:03
%S A282775 1,6,28,120,228,496,672,8128,30240,32760,125640,501888,523776,1207944,
%T A282775 2178540,23569920,29720448,33550336,45532800,142990848,459818240,
%U A282775 1379454720,1476304896,8589869056,14182439040,31998395520,43861478400,51001180160
%N 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.
%C A282775 The multiply-perfect numbers are a subset.
%C A282775 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.
%C A282775 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, ...
%C A282775 a(29) > 6 * 10^10. - _Lucas A. Brown_, Mar 10 2021
%H A282775 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A282774%2B5.py">A282774+5.py</a>
%e A282775 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.
%p A282775 with(numtheory): P:=proc(q) local a,b,c,k,n;
%p A282775 for n from 1 to q do if not isprime(n) then a:=sigma(n); b:=sort([op(divisors(n))]);
%p A282775 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);
%o A282775 (PARI) isok(k) = if (!isprime(k), my(sk = sigma(k)); (sk - sumdiv(k, d, sk % d)) % k == 0;); \\ _Michel Marcus_, Jun 17 2017
%Y A282775 Cf. A000203, A007691, A282774.
%K A282775 nonn
%O A282775 1,2
%A A282775 _Paolo P. Lava_, Feb 22 2017
%E A282775 a(16)-a(24) from _Giovanni Resta_, Feb 23 2017
%E A282775 a(25)-a(28) from _Lucas A. Brown_, Mar 10 2021