A127699 Length of period of the sequence (1^1^1^..., 2^2^2^..., 3^3^3^..., 4^4^4^..., ...) modulo n.

1, 2, 6, 4, 20, 6, 42, 8, 18, 20, 220, 12, 156, 42, 60, 16, 272, 18, 342, 20, 42, 220, 5060, 24, 100, 156, 54, 84, 2436, 60, 1860, 32, 660, 272, 420, 36, 1332, 342, 156, 40, 1640, 42, 1806, 220, 180, 5060, 237820, 48, 294, 100, 816, 156, 8268, 54, 220, 168

Offset

1,2

Comments

For any positive integers a and m the sequence a, a^a, a^a^a, a^a^a^a,... becomes eventually constant modulo m. So the remainder of a^a^a^... modulo n is well-defined.

Shapiro and Shapiro treat this problem. - T. D. Noe, Jan 30 2009

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

Daniel B. Shapiro and S. David Shapiro, Iterated Exponents in Number Theory, Integers 7 (2007), #A23.

Formula

a(n) = lcm(n, a(lambda(n))), where lambda is Carmichael's reduced totient function. - T. D. Noe, Jan 30 2009

Example

a(10)=20 because the last digit of 1^1^1^.. is 1; the sequence 2,2^2,2^2^2,.. ends with 2,4,6,6,...; the sequence 3,3^3,3^3^3,... with 3,7,7,...; 4,4^4,4^4^4,... with 4,6,6,...; and so on. We get as last digits 1,6,7,6,5,6,3,6,9,0, 1,6,3,6,5,6,7,6,9,0 and then the pattern repeats.

Maple

a:= proc(n) option remember; `if`(n=1, 1,
      ilcm(n, a(numtheory[lambda](n))))
    end:
seq(a(n), n=1..56);  # Alois P. Heinz, Jan 03 2023

Mathematica

nn=100; a=Table[0, {nn}]; a[[1]]=1; Do[a[[n]]=LCM[n, a[[CarmichaelLambda[n]]]], {n, 2, nn}]; a (* T. D. Noe, Jan 30 2009 *)

Prog

(Python)
from functools import lru_cache
from math import lcm
from sympy import reduced_totient
@lru_cache(maxsize=None)
def A127699(n): return 1 if n == 1 else lcm(n, A127699(reduced_totient(n))) # Chai Wah Wu, Jan 03 2023

Crossrefs

Cf. A002322.

Sequence in context: A275121 A174824 A009262 * A100140 A220769 A124838

Adjacent sequences: A127696 A127697 A127698 * A127700 A127701 A127702

Keyword

easy,nonn

Author

Jan Fricke, Apr 11 2007

Extensions

Extension and correction from T. D. Noe, Jan 30 2009

Incorrect formula removed by T. D. Noe, Feb 02 2009

Status

approved