[seqfan] n/n-1 Multiply Unitary[ Perfect Number
zbi74583.boat at orange.zero.jp
zbi74583.boat at orange.zero.jp
Thu Mar 5 06:23:52 CET 2009
Multiply Perfect Number is defined as follows.
Sigma(m)=n*m
If we replace Sigma as UnitarySigma then it is Multiply Unitary Perfect Number.
UnitarySigma(m)=n*m
But no exanple exists for 2<n.
I considered the following number.
UnitarySigma(m)=n/(n-1)*m
It has many examples.
I give the smallest examples for each n.
The case of n-1=p^i is easy, because UnitarySigma(p^i)=p^i+1
I have found no example for n=22, 23.
It is unsolved probrem.
n=3
m=2
n=4
m=3
n=5
m=2^2
n=6
m=5
n=7
m=2^3*3^3
n=8
m=7
n=9
m=2^3
n=10
m=3^2
n=11
m=2^5*3^3*5^3*7^2
n=12
m=11
n=13
m=2^9*3^6*5^3*7^2*19^2*73*181
n=14
m=13
n=15
m=2^10*5^4*7^2*41*79*157*313
n=16
m=3^3*5^3*7^2
n=17
m=2^4
n=18
m=17
n=19
m=2^9*3^5*31*61
n=20
m=19
n=21
m=2^10*5^4*41*79*157*313
Yasutoshi
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