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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, 1, 3, 12, 83, 1323, 63090, 14736464
OFFSET
0,3
COMMENTS
Number of ways any integer is a product of n superparticular ratios, without regard to order. A superparticular ratio is a ratio of the form (m+1)/m.
EXAMPLE
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.
CROSSREFS
KEYWORD
hard,nonn,more
AUTHOR
Keith F. Lynch, Sep 05 2024
STATUS
approved