[seqfan] n/n-1 Multiply Unitary[ Perfect Number

[zbi74583.boat at orange.zero.jp]

    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