UO-Sigma

[y.kohmoto]

    Hello, Seqfans.
    I posted sequences related a mixed divisor function on OEIS.
    http://www.research.att.com/projects/OEIS?Anum=A091321

    Naturally I considered the reverse of OU-sigma function.


    a(m) : 2*3, 2^2*3*5, 2^3*3^3*5, 2^3*3^2*7*13, 2^4*3^3*5*17,
2^4*3^2*7*13*17, 2^5*3^3*5*11, 2^5*3^2*7*13*11, 2^6*3*5*7*13,
2^6*3^2*5*7*13^2*31*61, 2^7*3^2*7*11*13*43, 2^7*3^3*5*11*43,
2^8*3^2*7*11*13*43*257, 2^8*3^3*5*11*43*257, 2^9*3^4*7*11^2*19^2*127,
2^9*3^4*7*11^2*19^4*151*911, 2^9*3^5*7^2*13*19^2*127,
2^9*3^5*7^2*13*19^4*151*911, 2^10*3*5^2*7*31*41,
2^11*3^4*7*11^2*19^2*127*683, 2^11*3^4*7*11^2*19^4*151*683*911,
2^11*3^5*7^2*13*19^2*127*683, 2^11*3^5*7^2*13*19^4*151*683*911,

    b(m) : 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
　　

    c(m) :    2*3, 2^3*3^2*7*13, 2^5*3^2*7*13*11, 2^7*3^2*7*11*13*43,
2^8*3^2*7*11*13*43*257,

    a(m) :         UO-Sigma(n)=k*n , for some iinteger k.
                     If n=Product p_i^r_i then UO-Sigma(n)=
UnitarySigma(2^r_1)*Sigma(n/2^r_1)
                        =(2^r_1+1)*Product(p_i^(r_i+1)-1)/(p_i-1) , 2<p_i

    example :    UO-Sigma(2^4*7^2)=UnitarySigma(2^4)*Sigma(7^2)=17*57= 969
                     So,  UO-Sigma(n) = UnitarySigma(n)        if n=2^r
                                              = Sigma(n)                  if
GCD(2,n)=1

    b(m) :         k values of a(m).
                     It seems to be a constant sequence, but I conjectured
that any positive integer more than 2 must appear.

                     I feel k=3 will appear at around 50th term.
                     The term whose k is four will have  about 100 digits.
                     Because k values of UO-PN corresponds 1/2*{k values of
Multiple PN}
                     The smallest 8ple PN has   133  digits.

    c(m) :         Some interesting part-sequences of a(n) exist.
                     It has a character that c(n-1)|c(n).

    comment :   the terms which has a form 2^9*k correspond the terms form
2^11*k' .
                     their factors are almost same each other.

    I will post them soon.

    Yasutoshi