A122221 Largest number k such that k! < (n!)^n.
2, 5, 8, 13, 19, 25, 32, 41, 50, 60, 72, 84, 97, 111, 126, 142, 159, 177, 196, 216, 237, 259, 282, 306, 330, 356, 383, 410, 439, 469, 499, 531, 563, 597, 631, 667, 703, 740, 779, 818, 858, 899, 942, 985, 1029, 1074, 1120, 1167, 1215, 1264, 1314, 1365, 1417
Offset: 2
Examples
a(3)=5 because 5! = 120 is less than (3!)^3 = 216 whereas 6! = 720 > 216.
Links
- Chai Wah Wu, Table of n, a(n) for n = 2..10000
Programs
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Maple
a:=proc(n) local b: b:=proc(k) if k!<(n!)^n then k else fi end: max(seq(b(k),k=1..2200)) end: seq(a(n),n=2..67); # Emeric Deutsch, Oct 07 2006
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Mathematica
s={};Do[k=1;Until[k!>=(n!)^n,k++]; AppendTo[s,k-1],{n,2,54}];s (* James C. McMahon, Oct 26 2024 *)
Formula
From Stirling's approximation, a(n) ~ n^2/2. A closer approximation for a(n) is n^2/2-c*n^2/log(n), where c = (1+log(0.5))/4 = A382854/2. - Johann Peters, Aug 23 2025
Extensions
More terms from Emeric Deutsch, Oct 07 2006
Comments