[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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