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A317824 revision #74

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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".

REFERENCES

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

LINKS

Table of n, a(n) for n=1..19.

M. Ripà, On the Convergence Speed of Tetration, ResearchGate (2018).

Wikipedia, Tetration

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.

EXAMPLE

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).

CROSSREFS

Cf. A000422, A058183, A171882 (tetration).