A255344 Product_{k=1..n} k^(k^5).

1, 4294967296

Offset: 1

Author

Vaclav Kotesovec, Feb 21 2015

Keywords

• nonn

Comments

The next terms: a(3) has 126 digits, a(4) has 743 digits, a(5) has 2927 digits.

In general, product_{k=1..n} k^(k^m) ~ A(m) * n^(B(m+1)/(m+1) + sum_{j=1..n} j^m) * exp(-n^(m+1)/(m+1)^2 + sum_{j=1..m-1} (1/(j+1) * B(j+1) * binomial(m,j) * n^(m-j) * sum_{i=0..j-1} 1/(m-i) )), where A(m) is the generalized Glaisher-Kinkelin constant (see A074962, A243262, A243263, A243264, A243265), and B(n) is the Bernoulli number A027641(n) / A027642(n).

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..4

Crossrefs

Cf. A002109, A051675, A255321, A255323, A243265.

Programs

Mathematica

Table[Product[k^(k^5), {k, 1, n}], {n, 1, 5}]
FoldList[Times,Table[k^k^5,{k,5}]] (* Harvey P. Dale, Aug 05 2025 *)

PARI

a(n)=prod(k=2,n,k^k^5) \\ Charles R Greathouse IV, Sep 08 2015

Formula

a(n) ~ A243265 * n^(n^2*(n+1)^2*(2*n^2+2*n-1)/12 + 1/252) / exp(47*n^2/720 - n^4/12 + n^6/36).