A255433 a(n) = Product_{k=0..n} (k^3+1).

1, 2, 18, 504, 32760, 4127760, 895723920, 308129028480, 158070191610240, 115391239875475200, 115506631115350675200, 153854832645647099366400, 266015005644323834804505600, 584700982406223788900303308800, 1605004196705084300531332582656000

Offset: 0

Author

Vaclav Kotesovec, Feb 23 2015

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..150
• Erhan Gürela, Ali Ulas Özgür Kisisel, A note on the products (1^mu + 1)(2^mu + 1)···(n^mu + 1), Journal of Number Theory, Volume 130, Issue 1, January 2010, Pages 187-191.
• Chuan Ze Niu, (1^3+1)(2^3+1)...(n^3+1) is not a cube, arXiv:1612.08158 [math.NT], 2016.

Crossrefs

Cf. A101686, A158621, A255434, A255435.

Programs

Mathematica

Table[Product[k^3 + 1, {k, 0, n}], {n, 0, 20}]
FullSimplify[Table[(Cosh[Sqrt[3]*Pi/2] * Gamma[2+n] * Gamma[1/2 - I*Sqrt[3]/2 + n] * Gamma[1/2 + I*Sqrt[3]/2 + n])/Pi, {n, 0, 20}]]
FoldList[Times,Range[0,20]^3+1] (* Harvey P. Dale, Jul 07 2017 *)

PARI

a(n) = prod(k=1, n, 1+k^3); \\ Michel Marcus, Jan 25 2016

Formula

a(n) ~ 2*sqrt(2*Pi) * cosh(sqrt(3)*Pi/2) * n^(3*n + 3/2) / exp(3*n).

a(n) = 2*A158621(n). - Vaclav Kotesovec, Jul 11 2015