A370601 a(n) is the number of integer values of the function F_2n(x) = Product_{i=0..2n-1} (x + i) / Sum_{i=0..2n-1} (x + i).

0, 1, 3, 23, 37, 156, 371, 1207, 2826, 8738, 14839, 60738, 80177, 148702, 466545, 1673039, 2077633, 4771287, 10665251, 26790730, 72170979, 212182718, 248771227, 1074691776, 1488526850, 2533202074, 8444006973, 13950326222, 18313391221, 74263958970, 164820521219

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

Table of n, a(n) for n=1..31.

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

Cf. A000005, A001818, A126695.

Sequence in context: A006598 A245623 A106892 * A116893 A106066 A281551

Adjacent sequences: A370598 A370599 A370600 * A370602 A370603 A370604

Keyword

nonn

Author

Vladimir Letsko, Feb 23 2024

Status

approved