cp's OEIS Frontend

A320523 Smallest m > 1 such that either n^m == n (mod 25) or n^m == 0 (mod 25).

%I A320523 #36 Dec 18 2018 02:27:12
%S A320523 2,21,21,11,2,6,5,21,11,2,6,21,21,11,2,6,21,5,11,2,6,21,21,3,2,2,21,
%T A320523 21,11,2,6,5,21,11,2,6,21,21,11,2,6,21,5,11,2,6,21,21,3,2,2,21,21,11,
%U A320523 2,6,5,21,11,2,6,21,21,11,2,6,21,5,11,2,6,21,21,3,2
%N A320523 Smallest m > 1 such that either n^m == n (mod 25) or n^m == 0 (mod 25).
%C A320523 This is a periodic sequence. In fact, a(n) (mod 25) == a(n + k*25) (mod 25), for any k >= 0. The maximum value of a(n) is 21 = lambda(25) + 1 = 20 + 1, since 20 is the Carmichael's lambda value in 25.
%C A320523 This sequence, omitting a(n = 10*k), predicts the convergence speed of any tetration a^^b, for any b >= a > 2, since A317905(n) = 1 iff a(n) > 5 and A317905(n) >= 2 otherwise (for any 2 <= a(n) <= 5).
%D A320523 M. Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6.
%H A320523 J. Jimenez Urroz and J. Luis A. Yebra, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Yebra/yebra4.html">On the equation a^x == x (mod b^n)</a>, J. Int. Seq. 12 (2009) #09.8.8.
%H A320523 M. Ripà, <a href="https://www.researchgate.net/publication/328493277_On_the_Convergence_Speed_of_Tetration">On the Convergence Speed of Tetration</a>, ResearchGate (2018).
%H A320523 M. Ripà, <a href="http://vixra.org/abs/1810.0223">On the Convergence Speed of Tetration</a>, viXra (2018).
%H A320523 Wikipedia, <a href="https://en.wikipedia.org/wiki/Carmichael_function">Charmichael function</a>
%H A320523 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%H A320523 Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function">Euler's totient function</a>
%F A320523 For any k >= 0,
%F A320523   a( 1 + k*25) =  2;
%F A320523   a( 2 + k*25) = 21;
%F A320523   a( 3 + k*25) = 21;
%F A320523   a( 4 + k*25) = 11;
%F A320523   a( 5 + k*25) =  2;
%F A320523   a( 6 + k*25) =  6;
%F A320523   a( 7 + k*25) =  5;
%F A320523   a( 8 + k*25) = 21;
%F A320523   a( 9 + k*25) = 11;
%F A320523   a(10 + k*25) =  2;
%F A320523   a(11 + k*25) =  6;
%F A320523   a(12 + k*25) = 21;
%F A320523   a(13 + k*25) = 21;
%F A320523   a(14 + k*25) = 11;
%F A320523   a(15 + k*25) =  2;
%F A320523   a(16 + k*25) =  6;
%F A320523   a(17 + k*25) = 21;
%F A320523   a(18 + k*25) =  5;
%F A320523   a(19 + k*25) = 11;
%F A320523   a(20 + k*25) =  2;
%F A320523   a(21 + k*25) =  6;
%F A320523   a(22 + k*25) = 21;
%F A320523   a(23 + k*25) = 21;
%F A320523   a(24 + k*25) =  3;
%F A320523   a(25*(k + 1))=  2.
%e A320523 For n = 41, a(41) = a(16) = 6, since 16^6 mod 25 = 16.
%t A320523 With[{k = 25}, Table[If[Mod[n, 5] == 0, 2, SelectFirst[Range[2, CarmichaelLambda@ k + 1], PowerMod[n, #, k] == Mod[n, k] &]], {n, 75}]] (* _Michael De Vlieger_, Oct 15 2018 *)
%o A320523 (PARI) a(n) = {my(m=2); while ((Mod(n, 25)^m != n) && (Mod(n, 25)^m != 0), m++); m;} \\ _Michel Marcus_, Oct 16 2018
%Y A320523 Cf. A067251, A317905.
%K A320523 nonn,easy
%O A320523 1,1
%A A320523 _Marco Ripà_, Oct 14 2018