# [seqfan] m/n-Unitary SN

zbi74583.boat at orange.zero.jp zbi74583.boat at orange.zero.jp
Wed Feb 19 06:17:59 CET 2014

```    Hi,Seqfans
[1/m-Aliquot Sequence]

Definiition
a(n)=1/m*Sigma(a(n-1))
Where  m is integer

Three cases exist.
1. Periodic
2. It becomes non integer
3. Divergent

The first one is 1/m-Sociable Number of order k.
a(n+k)=a(n), k is number of the members of a cycle.
m-Multipry Perfect Number is the fixed point of mapping E0.

A113546, A113791, A113285

I found one more  example of 1/4-SN

6943520030720, 5386209454080, 6389772480000, 7665533854902, 575659229184,
5657834203560, 6382469882880, 7938550287360, 9055131471360, 9256155068160,
11798498589696, 12316979136000, 13937317954560, 18314063179776,
18071888486400, 20136798781440, 18850467643392, 16492422758400,
15699547839573, 6635615891520, 7400619883008, 6494644316160, 8009835724800,
9317732578770, 7847237806128

Could anyone confirm it?

Recently I computed by hand the case of Unitary Sigma.

[m/n-Unitary Aliquot Sequence]
a(k)=m/n*UnitarySigma(a(k-1)) .... E1
Where m,n are integer, m<n

If it becomes periodic then it is m/n-Unitary Sociable Number.
Note that m/n-Unitary RMPN is the fixed point of mapping E1.

[Results]
2/3-Unitary SN

2^3, 2*3
2^2*3*5, 2^4*5, 2^2*17

3/4-Unitary SN

3^3, 3*7, 2^3*3
3^4*5, 3^2*41, 3^2*5*7, 2^3*3^2*5
2^3*3^4*5, 3^4*41, 3^2*7*41, 2^3*3^2*5*7
2^3*3^4*5*7, 2^3*3^4*41, 3^4*7*41, 2^3*3^2*7*41

Yasutoshi

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