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

1,2

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

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

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000312, A009262, A127699.

Cf. A002322, A124240, A268336.

Sequence in context: A066092 A100695 A275121 * A009262 A127699 A100140

Adjacent sequences: A174821 A174822 A174823 * A174825 A174826 A174827

Keyword

nonn

Author

Franklin T. Adams-Watters, Mar 30 2010

Status

approved