A317824 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))).

1, 21, 721, 8721, 8721, 708721, 5708721, 65708721, 165708721, 65165708721, 1165165708721, 861165165708721, 5861165165708721, 5005861165165708721, 55005861165165708721, 48055005861165165708721, 8448055005861165165708721, 388448055005861165165708721, 49388448055005861165165708721

Offset: 1

Author

Marco Ripà, Aug 10 2018

Keywords

• nonn
• base

Comments

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".

Examples

			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).
		

References

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

Links

• Marco Ripà, On the Convergence Speed of Tetration, ResearchGate (2018).
• Wikipedia, Tetration

Crossrefs

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

Programs

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))));}
f(n) = my(t=n); forstep(k=n-1, 1, -1, t=t*10^#Str(k)+k); t; \\ A000422
a(n) = my(x=f(n)); tmod(x, 10^#Str(x)); \\ Michel Marcus, Sep 12 2021

Formula

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.

Extensions

More terms from Jinyuan Wang, Aug 30 2020