fixed point

[y.kohmoto]

    Jud McCranie wrote :


    >At 04:13 AM 7/19/2003, you wrote:

    >>Hello seqfans.
    I am going to post the following two sequences to OEIS.

    [ seq.1. ]
    Fixed points of a mapping such that usigma(uphi(n)) :

        usigma(uphi(n)) = n        ....(1)

    It is easy to prove the following formula satisfies the equation 1.

    (2^r)^i*(F_m)^j*(3^2)^k , where 2^r-1=M_r , M_r is Mersenne Prime , F_m
is Fermat Prime , i=0 or 1 , j=0 or 1 , k=0 or 1 , j*k is not 1 .

    I thought different type of solutions exist.
    And I found one sporadic solution up to 10^7.

    44352 = 2^6*3^2*7*11

    [ seq.2. ]
    Fixed points of a mapping such that uphi(usigma(n)) :

    uphi(usigma(n)) = n             ....(2)

    A formula which satisfies 2  :

    (2^r)^i*(M_k)^j*(2^3)^s , where 2^r+1=F_t . F_t is Fermat Prime , M_k is
Mersenne Prime, i=0 or 1 , j=0 or 1 , s=0 or 1 , i*s is not 1 .

    sporadic solutions up to 10^7 :

    30240 = 2^5*3^3*5*7
    40920 = 2^3*3*5*11*31

    I want to know if any more sporadic solutions for the equations 1 and 2
exist.

    My computer has a CPU 366 M Hz. It doesn't run so rapidly.
    If you have a faster computer, try to search them and tell me the
result.

    Yasutoshi
\

>For the first one I get:

3,4,5,8,9,12,17,20,24,32,36,40,68,72,96,128,136,160,257,288,384,544,
640,1028,1152,2056,2176,8192,8224,24576,32896,40960,44352,65537,
73728,131072,139264,262148,393216,524288,524296,655360,1179648,1572864

for the second one I get:

2,3,4,6,7,8,12,14,16,24,28,31,48,56,62,112,124,127,248,254,256,496,
508,768,1016,1792,2032,7936,8191,16382,30240,32512,32764,65528,65536,
131056,131071,196608,262142,458752,524284,524287,1048568,1048574

         -----------

    Dear Jud
    Thanks. But I think my English was not good.

    I want to know about the "sporadic" solutions which have more digits
than the examples which I found.
    I searched the solutions up to 10^7, and I got the same results as
yours.

    See carefully your  results, almost all are the solutions which are
represented with the formulas except one and two sporadic solution which I
found for each sequences.

    For example, the last ones are factorized as follows :
    1572864 = 3*2^19
    1048574 = 2*(2^19-1)
    both are the formula's type. which I wrote.

    Yasutoshi