[seqfan] SigmaFibo_p(n)

[zbi74583.boat at orange.zero.jp]

    Hi, Seqfans


    I mixed A078414 and A000458.

    S_p(n)=(Sigma(a(n-1))+Sigma(a(n-2)))/p^m
                                Where m is the highst power of p dividing
(Sigma(a(n-1))+Sigma(a(n-2)))

    In general,   if S(n) is exponentially increase then S_p(n) have both
exponentially increase and decrease part , so it becomes rather difficult to
predict the behavior of S_p(n).

    I computed several cases


    SigmaFibo_2(n) :


    1,1,1,1,....

    1,2,1,1,....

    1,3,5,5,3,5,....

    1,4,1,1,....

    1,5,7,7,1,9,7,21,5,19,13,17,1,19,21,13,23,19,11,1,13,15,19,11,....

    1,6,13,13,7,11,5,9,19,33,17,33,33,3,13,9,27,53,47,51,15,3,7,3,3,1,5,....

    1,7,9,21,45,55,75,49,181,239,211,113,163,139,19,5,13,5,5,....

    1,8,1,1,....

    1,9,7,21,5,19,13,17,1,19,21,13,23,19,11,1,13,15,19,11,....

    1,10,19,19,5,....

    These seem  to be periodic.


    SigmaFibo_3(n) :


    1,1,2,4,10,25,49,88,79,260,668,....
    1,1,3,7,18,31,57,180,80,588,.... Sigma(S_p(n))

    1,3,5,10,8,11,1,13,5,20,16,73,35,122,26,76,182,....
    1,4,6,18,15,12,1,14,6,42,31,74,48,186,42,140,.... Sigma(S_p(n))

    1,4,8,22,17,2,7,11,20,2,5,1,7,1,1,....
    1,7,15,36,18,3,8,12,42,3,6,1,8,1,1,.... Sigma(S_p(n))

    1,5,7,14,32,29,31,62,128,137,....
    1,6,8,24,63,30,32,96,255,.... Sigma(S_p(n))

    1,6,13,26,56,2,41,5,16,37,23,62,40,62,62,64,223,13,238,....
    1,12,14,42,120,3,42,6,31,38,24,96,90,96,96,127,224,14,.... Sigma(S_p(n))

    1,8,16,46,103,176,476,460,224,56,208,554,....
    1,15,31,72,104,372,1008,1008,504,120,434,.... Sigma(S_p(n))

    1,9,14,37,62,134,100,421,71,494,304,1460,....
    1,13,24,38,96,204,217,422,72,840,620,.... Sigma(S_p(n))

    1,10,19,38,80,82,104,112,458,938,....
    1,18,20,60,186,126,210,248,690,.... Sigma(S_p(n))

    1,8,16,46,103,176,476,....
    1,15,31,72,104,372,.... Sigma(S_p(n))


    Some of them seem  to be divergent.



    Yasutoshi