A362738 a(n) is the least nonnegative integer solution for y such that x > 1 is an integer in the equation n^y*x^n = n^(x^(1/n)).

0, 0, 192, 3000, 46440, 823200, 16776704, 387419760, 9999999000, 285311669280, 8916100446528, 302875106590056, 11112006825555272, 437893890380856000, 18446744073709547520, 827240261886336759264, 39346408075296537569592, 1978419655660313589117120, 104857599999999999999992000

Offset

2,3

Comments

Corresponding solutions for x are 256 (2^(2^3)) at n = 2, 19683 (3^(3^2)) at n = 3 and n^(n^2) for all n > 3.

Further solution pairs are of the form y = n^(n^k) - n^(2+k) and x = n^(n^(1+k)) with k > 0.

Links

Table of n, a(n) for n=2..20.

Formula

E.g.f.: 1/(1 + LambertW(-x)) - x*(1 + 3*x + x^2)*exp(x) + 2*x^2, where LambertW() is the Lambert W-function.

a(n) = n^n - n^3 for n > 2.

Prog

(PARI) a(n) = max(0, n^n-n^3)

Crossrefs

Cf. A000312, A058126, A073084.

Sequence in context: A351632 A174384 A232919 * A209594 A168635 A299863

Adjacent sequences: A362735 A362736 A362737 * A362739 A362740 A362741

Keyword

nonn,easy

Author

Thomas Scheuerle, May 01 2023

Status

approved