A051675 "Second factorials": Product_{k=1..n} k^(k^2).

1, 16, 314928, 1352605460594688, 403107840000000000000000000000000, 4157825501361955044460594652554199040000000000000000000000000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• Spyros G. Kanellos: Mathematical Researches and Results [in Greek]. Athens, 1965.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..13
• Karim Belabas and Henri Cohen, Numerical Algorithms for Number Theory Using PariGP, American Mathematical Society, 2021, p. 196.

Crossrefs

Cf. A002109, A255321, A255323, A255344.

Cf. A243262.

Programs

Magma

[(&*[k^(k^2): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Maple

A051675 := proc(n) local k; mul(k^(k^2),k=1..n); end;

Mathematica

Table[Product[k^(k^2),{k,1,n}],{n,1,10}] (* Vaclav Kotesovec, Feb 20 2015 *)

PARI

for(n=1, 10, print1(prod(k=1,n, k^(k^2)), ", ")) \\ G. C. Greubel, Oct 14 2018

Formula

a(n) ~ n^(n*(n+1)*(2n+1)/6) / exp(n^3/9 - n/12 - Zeta(3)/(4*Pi^2)), where Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 20 2015