[seqfan] PF Sequence

[zbi74583.boat at orange.zero.jp]

    Hi, Seqfans
    I found an intersting sequence related with divisor function.

    [Definition of PrimeFactor_p or PF_p]

         PrimeFactor_p[n] = The largest power of p diiding n
         It is written as PF_p[n]


         PF_p,q,r[n] = PF_p[n]* PF_q[n]* PF_r[n]
         PF_1[n] = 1

    [How to compute a(m)]
    Case of Base Primes = {2}{3,5}

    A=2^m, B=2^m
    Do[
         A=A/PF_2[UnitarySigma[B]]*PF_3,5[UnitarySigma[B]]
         B=UnitarySigma[B]/ PF_2,3,5[UnitarySigma[B]]

         IF B=1 THEN END   ]

    a(m) = A

    [Example]
    m=5
    A=2^5 , B=2^5
    T=1
         A=2^5/PF_2[3*11]* PF_3,5[3*11] =2^5*3
         B=3*11/ PF_2,3,5[3*11] =11

    T=2
         A=2^5*3/PF_2[2^2*3]* PF_3,5[2^2*3] =2^3*3^2
         B=2^2*3/ PF_2,3,5[2^2*3] = 1

    a(5)=2^3`3^2

         Base Prime={2}{3,5} UnitarySigma
         b1(m) : 1, 2*3, 2^2*5, 2^3*3^2, 2^3*3^2, 2^3*3^2, 2^2*5, 2^3*3^2,
2^3*3^2, 2^7*3^3*5, 2^4*3*5^2, 2^7*3^3*5, 2^3*3^2

         Base Prime={2}[3}
         b2(m) : 1, 6, 6, 72, 72, 72, 6, 72, 72, 5184, 6, 5184, 72

         Base Prime={2}{1}
         b3(m) : 1, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/4

         I don't understand well what the difference between m<14 2^m and 2^14 is.

    [How to compute a(m)  more generalized case]
    F(n) means divisor function such as Sigma or UnitaryPhi or (-1)Sigma etc.

    Case of Base Primes = {p}{q_i} , 1<=i<=k

    A=p^m, B=p^m
    Do[
         A=A/PF_p[F [B]]*PF_{q_i , 1<=i<=k } [F [B]]
         B=F [B]/ PF_{p, q_i , 1<=i<=k } [F [B]]

         IF B=1 THEN END   ]

    a(m) = A

         Base Prime = {2}{1} UnitaryPhi
         c3(m) : 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2

         Base Prime = {2}{3}
         c2(m) : 1, 2, 12, 12, 12, 12, 2^5*3^3, 2^5*3^3, 12, 2^5*3^3, 2^5*3^3

         Base Prime = {2}{3,5}
         c1(m) : 1, 2, 2^2*3, 2^2*3, 2^4*3*5, 2^4*3*5, 2^5*3^3, 2^5*3^3,
2^4*3*5, 2^5*3^3, 2^8*3^2*5^2

         Base Prime = {2}{3,7}
         c0(m) : 1, 2, 2^2*3, 2^2*3, 2^2*3, 2^2*3, 2^6*3^2*7, 2^-*3^2*7, 2^2*3,
2^6*3^2*7, 2^5*3^3

         Base Prime = {2}{1} Sigma
         d3(m) : 1, 1/2, 1/2, 1/4, 1/2, 1/4, 1/2, 1/8, 1/4, 1/2, 1/8

         Base Prime = {2}{3} Sigma
         d3(m) : 1, 2*3, 1/2, 2^2*3^2, 1/2, 2^2*3^2, 1/2, 2^5*3^4, 2^2*3^2,
2^2*3^2, 2^5*3^4

         Base Prime = {3}{2} UnitarySigma
         ｂ4(m) : 1, 2^2*3, 2^2*3, 2^5*3^3, 2^5*3^3, 2^8* ^5, 2^7*3^4

         Base Prime = {2}{1} (-1) Sigma
         f3(m) : 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

         Base Prime = {2}{3} (-1) Sigma
         f2(m) : 1, 2, 1,2*3, 2*3, 2*3, 2*3, 1, 2^4*3^3, 2*3, 2^4*3^3



    Yasutoshi