fixed point

[y.kohmoto]

    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