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Numbers m such that Sum_{d|m: 1<=d<=sqrt(m)} m/d-d is a multiple of m.
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%I #31 Mar 24 2021 12:36:51

%S 1,6,840,3420

%N Numbers m such that Sum_{d|m: 1<=d<=sqrt(m)} m/d-d is a multiple of m.

%C Integers m such that A079667(m) is a multiple of m.

%C Sum_{d|m: 1<=d<=sqrt(m)} (m/d)-d = 0 only when m=1 and Sum_{d|m: 1<=d<=sqrt(m)} (m/d)-d = m only when m=6.

%C If m is q+1-perfect and 2*Sum_{d|m: 1<=d<=sqrt(m)} d=m then Sum_{d|m: 1<=d<=sqrt(m)} (m/d)-d = m*q or if m is member of this sequence i.e. Sum_{d|m: 1<=d<=sqrt(m)} (m/d)-d = m*q and 2*Sum_{d|m: 1<=d<=sqrt(m)} d=m then m is q+1-perfect.

%C Does there exist any m apart from 6 which is q+1-perfect, q>=2 and satisfies 2*Sum_{d|m: 1<=d<=sqrt(m)} d=m? Because if it exists then m is member of this sequence.

%t Select[Range[10^5], Function[m, Mod[DivisorSum[m, Abs[m/# - #] &, # <= Sqrt[m] &], m] == 0]] (* _Michael De Vlieger_, Mar 17 2021 *)

%o (PARI) isok(m) = !(sumdiv(m, d, if (d^2 <= m, m/d-d)) % m); \\ _Michel Marcus_, Jan 28 2021

%Y Cf. A079667, A007691 (in comment).

%K nonn,more

%O 1,2

%A _Devansh Singh_, Jan 28 2021