A171882 -id:A171882 - cp's OEIS frontend

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

Marco Ripà, Aug 10 2018

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".

			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).

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

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

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

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

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.

More terms from Jinyuan Wang, Aug 30 2020

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))).

Original entry on oeis.org

4, 76, 176, 4176, 314176, 91314176, 891314176, 80891314176, 88080891314176, 5288080891314176, 705288080891314176, 10705288080891314176, 2410705288080891314176, 912410705288080891314176, 42912410705288080891314176, 9242912410705288080891314176, 989242912410705288080891314176

Offset: 1

Marco Ripà, Aug 10 2018

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".

			For n = 6, a(6) = 13117532^^13117532 (mod 10^8) == 91314176.

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

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

Cf. A038394, A068670, A171882 (tetration), A317824.

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) = fromdigits(concat([digits(p) | p<-Vecrev(primes(n))])); \\ A038394
a(n) = if (n==1, 4, my(x=f(n)); tmod(x, 10^#Str(x))); \\ Michel Marcus, Sep 12 2021

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.

More terms from Jinyuan Wang, Aug 30 2020

A321312 A(n,k) = n^^k is the k-th tetration of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 4, 3, 1, 0, 1, 16, 27, 4, 1, 1, 1, 65536, 7625597484987, 256, 5, 1

Offset: 0

Alois P. Heinz, Nov 03 2018

			Square array A(n,k) begins:
  1, 0,      1,              0,      1,   0,   1, ...
  1, 1,      1,              1,      1,   1,   1, ...
  1, 2,      4,             16,  65536, ...
  1, 3,     27,  7625597484987,    ...
  1, 4,    256,            ...
  1, 5,   3125,            ...
  1, 6,  46656,            ...
  1, 7, 823543,            ...
  ...

Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration
Index entries for sequences related to tetration

Columns k=0-3 give: A000012, A001477, A000312, A002488.
Rows n=0-4 give: A059841, A000012, A014221, A014222(k+1), A114561(k+1).
Main diagonal gives A004231 (Ackermann's sequence).
Cf. A027747, A171882 (by upwards diagonals).

A:= (n, k)-> `if`(k=0, 1, n^A(n, k-1)):
seq(seq(A(n, d-n), n=0..d), d=0..6);

A171881 Square array, read by antidiagonals, where T(n,k)=n^^k for n>=0, k>=1.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 27, 16, 1, 1, 5, 256, 19683, 256, 1, 1, 6, 3125, 4294967296, 7625597484987, 65536, 1, 1, 7, 46656, 298023223876953125, 340282366920938463463374607431768211456

Offset: 0

Robert Munafo, Jan 21 2010

n^^k is defined the left-associative way: n^^2=n^n, n^^3=(n^n)^n=n^(n^2), n^^4=((n^n)^n)^n=n^(n^3), and in general n^^k=n^(n^(k-1)).

More terms on Munafo website.

			Array begins:
  0,1,1,1,1,1,...
  1,1,1,1,1,1,...
  2,4,16,256,65536,...
  3,27,19683,...
  4,256,4294967296,...
  5,3125,...
  6,46656,...

Robert Munafo, Hyper4 Iterated Exponential Function.

Cf. A171882.

T[n_, k_] := If[n == 0, Boole[k != 0], n^(n^k)]; Table[T[k, n - k], {n, 0, 7}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Oct 29 2021 *)

A373495 a(1) = 2; thereafter, a(n) = prime(n)^prime(n-1) (mod 10).

Original entry on oeis.org

2, 9, 5, 7, 1, 7, 7, 9, 7, 9, 1, 3, 1, 3, 3, 7, 9, 1, 7, 1, 7, 9, 7, 9, 7, 1, 3, 3, 9, 3, 7, 1, 3, 9, 9, 1, 3, 3, 3, 7, 9, 1, 1, 7, 7, 9, 1, 7, 3, 9, 3, 9, 1, 1, 3, 3, 9, 1, 3, 1, 3, 7, 7, 1, 7, 7, 1, 3, 7, 9, 3, 9, 3, 7, 9, 7, 9, 7, 1, 9, 9, 1, 1, 7, 9, 7, 9, 7, 1, 3, 3, 9, 3, 1, 9, 7, 9, 1, 3, 1, 7, 3, 3, 9, 1

Offset: 1

Robert G. Wilson v, Jun 06 2024

nonn

This sequence is not periodic.

			a(2) = 3^2 (mod 10) = 9.
a(3) = 5^3 (mod 10) = 5.

Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing, Redwood City, CA, 1991, p. 226-229.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Robert P. Munafo, Hypercalc - The Calculator That Doesn't Overflow.
Robert P. Munafo, Sequence A092188, Smallest Positive Integer M such that 2^3^4^5^...^N = M mod N.
Robert G. Wilson v, Mathematica coding for "SuperPowerMod" from Vardi.
Wolfram cloud Function Repository, PowerTowerMod.

Cf. A000040, A092188, A133612, A171881, A171882, A213013, A336111, A342176.

a[n_] := Switch[ Mod[ Prime[n], 10], 1, 1, 3, If[ Mod[ Prime[n -1], 4] == 1, 3, 7], 5, 5, 7, If[ Mod[ Prime[n -1], 4] == 1, 7, 3], 9, 9]; a[1] = 2; a[2] = 9; Array[a, 105]
Join[{2}, Map[PowerMod[#[[2]], #[[1]], 10] &, Partition[Prime[Range[100]], 2, 1]]] (* Paolo Xausa, Jul 14 2025 *)

a(n) = if(n<2, 2, lift(Mod(prime(n),10)^prime(n-1))) \\ Hugo Pfoertner, Jul 07 2024

a(n) = A078422(n-1) mod 10. - R. J. Mathar, Jul 14 2025

A371720 a(n) = m^^m mod 10^len(m), where m = A038399(n) and ^^ indicates tetration or hyper-4.

Original entry on oeis.org

1, 11, 811, 3811, 63811, 763811, 3763811, 5103763811, 515103763811, 19515103763811, 6819515103763811, 8146819515103763811, 3808146819515103763811, 7213808146819515103763811, 9807213808146819515103763811, 4939807213808146819515103763811

Offset: 1

Marco Ripà, Apr 04 2024

For any n, a(n) == a(n + 1) (mod 10^len(A038399(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, A061839, A092845, and A104759), as shown in Ripà's book "La strana coda della serie n^n^...^n".

Moreover, assuming n > 1, since A038399(n) is congruent to 11 (mod 20), the convergence speed of A038399(n)^^b (say, V(A038399(n), b) = {2, 1, 1, 1, ...}) is 2 at height 1 and becomes a unit value for any integer b > 1 (see Links). Hence, a(n) is given by A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))), and also by A038399(n)^^len(A038399(n)) (mod 10^len(A038399(n))) since A038399(n)^^len(A038399(n)) == A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))) holds for any n.

			a(8) is given by the rightmost 10 digits of 2113853211^^2113853211 and thus a(8) = 5103763811.
a(9) == a(8) (mod 10^10), i.e., the digits of a(9) end with the digits of a(8) (and then a(9) has 2 more preceding).

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

Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441-457.
Wikipedia, Tetration.

Cf. A000045 (Fibonacci), A038399, A171882 (tetration), A317824, A317903, A317905.

a(n) = A038399(n)^^(len(A038399(n)) - 1) mod 10^len(A038399(n)), where len(A038399(n)) = ceiling(log_10(A038399(n) + 1)).

cp's OEIS Frontend

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))).

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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))).

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A321312 A(n,k) = n^^k is the k-th tetration of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

A171881 Square array, read by antidiagonals, where T(n,k)=n^^k for n>=0, k>=1.

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Mathematica

A373495 a(1) = 2; thereafter, a(n) = prime(n)^prime(n-1) (mod 10).

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A371720 a(n) = m^^m mod 10^len(m), where m = A038399(n) and ^^ indicates tetration or hyper-4.

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