A368723 a(n) = Product_{i=1..n, j=1..n, k=1..n} (i^4 + j^4 + k^4).

1, 3, 30180180096, 130911253854794147456410254996552949923277899497472

Offset: 0

Author

Vaclav Kotesovec, Jan 04 2024

Keywords

• nonn

Comments

Next term is too long to be included.

In general, for m>0, limit_{n->oo} (Product_{i=1..n, j=1..n, k=1..n} (i^m + j^m + k^m))^(1/(n^3)) / n^m = exp(Integral_{x=0..1, y=0..1, z=0..1} log(x^m + y^m + z^m) dz dy dx) = exp(Integral_{x=0..1, y=0..1} (log(1 + x^k + y^k) - k + k*hypergeom2F1(1/k, 1, (k+1)/k, -1/(x^k + y^k))) dy dx).

Crossrefs

Cf. A306594 (m=1), A324425 (m=2), A368722 (m=3).

Cf. A324437, A368721.

Programs

Mathematica

Table[Product[i^4 + j^4 + k^4, {i, 1, n}, {j, 1, n}, {k, 1, n}], {n, 0, 5}]

Formula

Limit_{n->oo} a(n)^(1/(n^3)) / n^4 = exp(Integral_{x=0..1, y=0..1, z=0..1} log(x^4 + y^4 + z^4) dz dy dx) = 0.3570458697635761757481417...