[seqfan] UnitaryPhi AP Generalized
zbi74583_boat at yahoo.co.jp
zbi74583_boat at yahoo.co.jp
Thu Dec 5 03:49:01 CET 2019
Hi Seqfans
Generally the definition of AP is described as follows
S(m) = S(n) = f(m, n) Where S is divisor function like Sigma, n-InfinitarySigma, Unitarysigma, UnitaryPhi etc f is function of m, n for instance, the easiest one is m + n
I computed more generalized UnitaryPhi AP
1. UnitaryPhi(x) = UnitaryPhi(y) = 2/(3*17^2)*(12*x^(1/2) + 5*y^(1/2))^3/(10*x^(1/2) + 7*y^(1/2))
If x = y then = 2/3*x
x(n) : 3 , 36 , 2^3*3^2*7 , 2^2*3^3*13 , 2^3*3^3*7*13 = A305678
x < y
5* 17^2*97*11^2 , 5*17^2*97*2^4*3^2
2.
UnitaryPhi(x) = UnitaryPhi(y) = 4/(5*7^2)*(6*x^(1/2) + y^(1/2))^2*(2*x^(1/2) + 5*y^(1/2))/(5*x^(1/2) + 2*y^(1/2))
If x = y then = 4/5*x
x(n) : 5 , 2^4*3^2*5^2 , 2^4*3^2*5^3*31
x < y
7^2*37*5^2 , 7^2*37*2^2*3^2
3.
UnitaryPhi(x) = UnitaryPhi(y) = 1/4*(x^(1/2) + y^(1/2))^3/(x^(1/2) + 2*y^(1/2))
If x = y then = 3/4*x
x(n) : 4 , 2^3*7 , 2^4*5 , 2^5*5*31 , 2^6*3^2*7 , 2^7*3^2*7*127
x < y
5*97*31^2, 5*97*3^2*11^2
4,
UniyaryPhi(x) = UnitaryPhi(y) = 4/(5*23^2)*(18*x^(1/2) + 5*y^(1/2))^3/(14*x^(1/2) + 9*y^(1/2))
If x = y then = 4/5*x
x(n) : 5 , 2^4*3^2*5^2 x < y 11*23^2*31*5^2, 11*23^2*31*2^2*3^2
I computed them mentally so I need that someone confirm them
Yasutoshi
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