A376012 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = any integer.
1, 1, 3, 12, 83, 1323, 63090, 14736464
Offset: 0
Examples
For n = 2, a(2) = 3, three solutions, {x_1, x_2} = {2, 3} = 2; {1, 2} = 3; {1, 1} = 4.
In other words, a(2) = 3 since 2 can be written as (3/2)(4/3), 3 can be written as (2/1)(3/2), and 4 can be written as (2/1)^2, but no other integers are the product of two superparticular ratios.
a(3) = 12 since 2 can be written in 5 ways, 3 can be written in 3 ways, and 4, 5, 6, and 8 can be written in 1 way each, as the product of three superparticular ratios.
Programs
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PARI
f(n, m, p, q)={ \\ the number of solutions in which product of the ratios is equal to p/q if(p<=q, return(0)); if(n==1, return(if(q%(p-q)==0, 1, 0))); x=floor(1/((p/q)^(1/n)-1)); \\ x is the maximum of the least denominator of the ratios my(ans=0); for(i=m, x, ans=ans+f(n-1, i, p*i, q*(i+1))); \\ m indicates that the solutions require the denominators of all ratios not to be less than m \\ discard the ratio (i+1)/i return(ans); }; a(n)={ my(ans=0); for(i=2, 2^n, ans=ans+f(n, 1, i, 1)); return(ans); }; \\ Yifan Xie, Nov 21 2024
Comments