A337162 Numbers m such that Sum_{d|m: 1<=d<=sqrt(m)} m/d-d is a multiple of m.

1, 6, 840, 3420

Offset: 1

Author

Devansh Singh, Jan 28 2021

Keywords

• nonn
• more

Comments

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

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.

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.

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.

Crossrefs

Cf. A079667, A007691 (in comment).

Programs

Mathematica

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

PARI

isok(m) = !(sumdiv(m, d, if (d^2 <= m, m/d-d)) % m); \\ Michel Marcus, Jan 28 2021