A174824 a(n) = period of the sequence {m^m, m >= 1} modulo n.

1, 2, 6, 4, 20, 6, 42, 8, 18, 20, 110, 12, 156, 42, 60, 16, 272, 18, 342, 20, 42, 110, 506, 24, 100, 156, 54, 84, 812, 60, 930, 32, 330, 272, 420, 36, 1332, 342, 156, 40, 1640, 42, 1806, 220, 180, 506, 2162, 48, 294, 100, 816, 156, 2756, 54, 220, 168, 342

Offset: 1

Author

Franklin T. Adams-Watters, Mar 30 2010

Keywords

• nonn

Comments

This is a divisibility sequence: if n divides m, a(n) divides a(m).

We have the equality n = a(n) for numbers n in A124240, which is related to Carmichael's function (A002322). The largest values of a(n) occur when n is prime, in which case a(n) = n*(n-1). - T. D. Noe, Feb 21 2014

Examples

			For n=3, 1^1 == 1 (mod 3), 2^2 == 1 (mod 3), 3^3 == 0 (mod 3), etc. The sequence of residues 1, 1, 0, 1, 2, 0, 1, 1, 0, ... has period 6, so a(3) = 6. - _Michael B. Porter_, Mar 13 2018
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• José María Grau and Antonio M. Oller-Marcén, On the last digit and the last non-zero digit of n^n in base b, arXiv:1203.4066 [math.NT], 2012. (See page 3)
• Index to divisibility sequences

Crossrefs

Cf. A000312, A009262, A127699.

Cf. A002322, A124240, A268336.

Programs

Mathematica

Table[LCM[n, CarmichaelLambda[n]], {n, 100}] (* T. D. Noe, Feb 20 2014 *)

PARI

a(n)=local(ps);ps=factor(n)[,1]~;for(k=1,#ps,n=lcm(n,ps[k]-1));n

PARI

a(n) = lcm(n, lcm(znstar(n)[2])); \\ Michel Marcus, Mar 18 2016; corrected by Michel Marcus, Nov 13 2019

PARI

apply( {A174824(n)=lcm(lcm([p-1|p<-factor(n)[,1]]),n)}, [1..99]) \\ [...] = znstar(n)[2], but 3x faster. - M. F. Hasler, Nov 13 2019

Formula

a(n) = lcm(n, A173614(n)) = lcm(n, A002322(n)) = lcm(n, A011773(n)).

If n and m are relatively prime, a(n*m) = lcm(a(n), a(m)); a(p^k) = (p-1)*p^k for p prime and k > 0.

a(n) = n*A268336(n). - M. F. Hasler, Nov 13 2019