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A317903 a(n) = A038394(n)^^A038394(n) (mod 10^len(A038394(n))), where ^^ indicates tetration or hyper-4 (e.g., 3^^4=3^(3^(3^3))).

%I A317903 #32 Sep 12 2021 12:17:24
%S A317903 4,76,176,4176,314176,91314176,891314176,80891314176,88080891314176,
%T A317903 5288080891314176,705288080891314176,10705288080891314176,
%U A317903 2410705288080891314176,912410705288080891314176,42912410705288080891314176,9242912410705288080891314176,989242912410705288080891314176
%N A317903 a(n) = A038394(n)^^A038394(n) (mod 10^len(A038394(n))), where ^^ indicates tetration or hyper-4 (e.g., 3^^4=3^(3^(3^3))).
%C A317903 For any n >= 2, a(n) (mod 10^len(A038394(n))) == a(n + 1) (mod 10^len(A038394(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 A014925 or A092447), as shown in Ripà's book "La strana coda della serie n^n^...^n".
%D A317903 Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, page 60. ISBN 978-88-6178-789-6
%H A317903 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).
%F A317903 a(n) = (p(n)_p(n-1)_p(n-2)_..._3_2)^^(p(n)_p(n-1)_p(n-2)_..._3_2) (mod 10^len(p(n)_p(n-1)_p(n-2)_..._3_2)), where len(k) := number of digits in k.
%e A317903 For n = 6, a(6) = 13117532^^13117532 (mod 10^8) == 91314176.
%o A317903 (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 A317903 f(n) = fromdigits(concat([digits(p) | p<-Vecrev(primes(n))])); \\ A038394
%o A317903 a(n) = if (n==1, 4, my(x=f(n)); tmod(x, 10^#Str(x))); \\ _Michel Marcus_, Sep 12 2021
%Y A317903 Cf. A038394, A068670, A171882 (tetration), A317824.
%K A317903 nonn,base
%O A317903 1,1
%A A317903 _Marco Ripà_, Aug 10 2018
%E A317903 More terms from _Jinyuan Wang_, Aug 30 2020