OFFSET
1,3
COMMENTS
If k is an odd prime then F_k(x) = Product_{i=0..k} (x + i)/ Sum_{i=0..k} (x + i) is not integer if and only if x==(k+1)/2 (mod k). If k is odd but not prime, then F_k(x) is integer for all positive integers x. On the other hand, for every even k, there is only a finite number of integer values of F_k(x).
LINKS
Vladimir Letsko, Math. vertical, YouTube video, 2023 (in Russian).
FORMULA
a(n) = tau(d/gcd(d, n)) - n, where d = ((2*n-1)!!)^2 and tau(k) is the number of divisors of the positive integer k.
EXAMPLE
a(3) = 3 because F_6(x) has exactly 3 integer values: F_6(5) = 3360, F_6(10) = 48048, and F_6(35) = 12282816.
MAPLE
a := proc(n) local d; d := doublefactorial(2*n-1)^2; numtheory[tau](d/igcd(n, d)) - n end: seq(a(n), n = 1..31);
MATHEMATICA
Table[DivisorSigma[0, (2*n - 1)!!^2/GCD[(2*n - 1)!!^2, n]] - n, {n, 1, 30}] (* Vaclav Kotesovec, Feb 23 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Letsko, Feb 23 2024
STATUS
approved