# [seqfan] Much easier generalized AP

zbi74583_boat at yahoo.co.jp zbi74583_boat at yahoo.co.jp
Sun Jan 26 05:44:20 CET 2020

```    Hi  Seqfans

My interest to AP have become pedantic    I feel no one compute such complicated formulas        UPhi(x) = UPhi(y) = k*(t*x^(1/2) + u*y^(1/2))^3/(v*x^(1/2) + w*y^(1/2))    t, u, v, w are integer    I am going to compute much easier generalized AP    I classified AP  because the difficulties of each are different
C.1              Original AP           S(m) = S(n) = m + n           S is divisor function
C.2           Linear AP
S(m) = S(n) = u*m + v*n           u, v are integer                                                                 u + v < 10
C 3           Rational AP           S(m) = S(n) = (m + n)^3/(m^2 + n^2)
http:// mathwor ld.wolf ram.com /Ration alAmica blePair .html
It is an example           In general ,     it  has rational formula
C 4           Irrational AP           S(m) = S(n) = 1/8*(5*m^(1/2) - 3*n^(1/2))^2
http:// oeis.or g/A1445 87
It is an example           In general ,     it has irrational formula
I     think the case of C.2 for small u, v should exist on OEIS    I computed the easiest generalized AP as follows           Sigma(x) = Sigma(y) = x + 2*y    First of all  I computed it mentally and found two terms           x(n) , y(n)  :  {2^9*3*31*5*13 , 2^9*3*31*83} , {2^13*3*127*239 , 2^13*3*127*7*29}    I conjectured the first one is the smallest and against my policy  I used PARI interactive computer    Program and the result is the following    prog(k) = {                          my(m =  90, u = 2, v = 1);   until(k<m, my(n =      (sigma(m) -   u*m)/v) ;                 if(0<n, if(1/u*(sigma(n) -   v*n) == m, print(factor(m), ",", factor(n)))); m++)};
prog(8000000)
[2, 3; 3, 1; 5, 1],[2, 3; 3, 1; 5, 1][2, 5; 3, 1; 7, 1],[2, 5; 3, 1; 7, 1][2, 2; 3, 1; 5, 1; 29, 1],[2, 3; 3, 1; 5, 1; 13, 1][2, 5; 3, 5],[2, 2; 3, 2; 5, 1; 41, 1][2, 2; 3, 2; 5, 1; 107, 1],[2, 5; 3, 2; 71, 1][2, 5; 3, 1; 5, 1; 43, 1],[2, 5; 3, 1; 263, 1][2, 3; 3, 5; 11, 1],[2, 5; 3, 2; 79, 1][2, 2; 3, 3; 7, 1; 29, 1],[2, 3; 3, 1; 7, 1; 139, 1][2, 2; 3, 2; 7, 1; 89, 1],[2, 3; 3, 2; 7, 1; 41, 1][2, 1; 3, 2; 5, 1; 7, 1; 41, 1],[2, 2; 3, 2; 7, 1; 107, 1][2, 4; 3, 1; 5, 1; 131, 1],[2, 9; 3, 1; 23, 1][2, 5; 3, 1; 11, 1; 53, 1],[2, 5; 3, 1; 5, 1; 107, 1][2, 5; 3, 2; 17, 1; 19, 1],[2, 3; 3, 2; 1511, 1][2, 1; 3, 1; 5, 1; 7, 1; 13, 1; 37, 1],[2, 2; 3, 1; 7, 1; 11, 1; 113, 1][2, 2; 3, 1; 5, 1; 29, 1; 61, 1],[2, 3; 3, 1; 5, 2; 167, 1][2, 1; 3, 3; 5, 1; 13, 1; 41, 1],[2, 2; 3, 3; 5, 1; 251, 1][2, 2; 3, 2; 13, 1; 17, 1; 19, 1],[2, 3; 3, 2; 13, 1; 167, 1][2, 3; 3, 2; 11, 1; 251, 1],[2, 5; 3, 2; 23, 1; 29, 1][2, 1; 3, 3; 5, 1; 11, 1; 83, 1],[2, 2; 3, 3; 5, 1; 431, 1][2, 1; 3, 5; 7, 2; 13, 1],[2, 2; 3, 2; 7, 2; 11, 1; 13, 1][2, 2; 3, 2; 13, 1; 17, 1; 41, 1],[2, 5; 3, 2; 13, 1; 83, 1][2, 1; 3, 2; 7, 1; 13, 1; 251, 1],[2, 2; 3, 2; 5, 1; 7, 1; 13, 1; 17, 1][2, 2; 3, 2; 7, 2; 239, 1],[2, 3; 3, 2; 7, 1; 797, 1][2, 5; 3, 2; 1487, 1],[2, 4; 3, 2; 5, 1; 503, 1][2, 3; 3, 2; 7, 1; 991, 1],[2, 4; 3, 1; 11, 1; 1039, 1][2, 9; 3, 1; 11, 1; 31, 1],[2, 9; 3, 1; 11, 1; 31, 1]    It is not all result  I will describe it on my blog    Notice the case of  m = n  is 3-Multiple PN    I recognized that computer is rather smart  because     it computed  many terms less than the one I conjectured the smallest

Yasutoshi

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