A379577 a(n) = (n!)^n + n^n.

2, 2, 8, 243, 332032, 24883203125, 139314069504046656, 82606411253903523840823543, 6984964247141514123629140377616777216, 109110688415571316480344899355894085582848387420489, 395940866122425193243875570782668457763038822400000000010000000000, 409933016554924328182440935903164918932547530146724293451448320000000000285311670611

Offset

0,1

Comments

The equation (k!)^n + k^n = (n!)^k + n^k holds if and only if n = k. See the Proof of the Theorem 2.2 (p.182-183) in the Alzer and Luca article in Links section.

Links

Table of n, a(n) for n=0..11.

Horst Alzer and Florian Luca, Diophantine equations involving factorials, Mathematica Bohemica 142.2 (2017), 181-184.

Formula

a(n) = A036740(n) + A000312(n).

Example

n = 3: a(3) = (3!)^3 + 3^3 = 243.

Maple

seq((n!)^n + n^n, n=0..12); # Georg Fischer, Dec 30 2024

Mathematica

a[n_]:=(n!)^n + n^n; Array[a, 12, 0] (* Stefano Spezia, Dec 26 2024 *)

Crossrefs

Cf. A000142, A000312, A036740.

Sequence in context: A180370 A326939 A341303 * A011148 A365307 A351400

Adjacent sequences: A379574 A379575 A379576 * A379578 A379579 A379580

Keyword

nonn,easy,new

Author

Ctibor O. Zizka, Dec 26 2024

Extensions

a(9)-a(11) corrected by Georg Fischer, Dec 30 2024

Status

approved