%I #79 Sep 12 2021 12:46:27

%S 1,21,721,8721,8721,708721,5708721,65708721,165708721,65165708721,

%T 1165165708721,861165165708721,5861165165708721,5005861165165708721,

%U 55005861165165708721,48055005861165165708721,8448055005861165165708721,388448055005861165165708721,49388448055005861165165708721

%N a(n) = A000422(n)^^A000422(n) (mod 10^len(A000422(n))), where ^^ indicates tetration or hyper-4 (e.g., 3^^4 = 3^(3^(3^3))).

%C For any n, a(n) (mod 10^len(A000422(n))) == a(n + 1) (mod 10^len(A000422(n))), where len(k) := number of digits in k. Assuming len(a(n))>1, this is a general property of every concatenated sequence with fixed rightmost digits (such as A061839 or A014925), as shown in Ripà's book "La strana coda della serie n^n^...^n".

%D Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, page 60. ISBN 978-88-6178-789-6

%H Marco 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>

%F a(n) = (n_n-1_n-2_..._2_1)^^(n_n-1_n-2_..._2_1) (mod 10^len(n_n-1_n-2_..._2_1)), where len(k) := number of digits in k.

%e For n = 3, a(3) = 321^^321 (mod 10^3) = 721. In fact, a(3) (mod 10^3) == a(4) (mod 10^3), since 721 (mod 10^3) == 8721 (mod 10^3).

%o (PARI) tmod(b, n) = {if (b % n == 0, return (0)); if (b % n == 1, return (1)); if (gcd(b, n)==1, return (lift(Mod(b, n)^tmod(b, lift(znorder(Mod(b, n))))))); lift(Mod(b, n)^(eulerphi(n) + tmod(b, eulerphi(n))));}

%o f(n) = my(t=n); forstep(k=n-1, 1, -1, t=t*10^#Str(k)+k); t; \\ A000422

%o a(n) = my(x=f(n)); tmod(x, 10^#Str(x)); \\ _Michel Marcus_, Sep 12 2021

%Y Cf. A000422, A058183, A171882 (tetration), A317903.

%K nonn,base

%O 1,2

%A _Marco Ripà_, Aug 10 2018

%E More terms from _Jinyuan Wang_, Aug 30 2020