A213013 -id:A213013 - cp's OEIS frontend

A092188 a(n) = smallest positive integer m such that 2^3^4^5^...^n == m (mod n).

2, 2, 4, 2, 2, 1, 8, 8, 2, 2, 8, 5, 8, 2, 16, 2, 8, 18, 12, 8, 2, 16, 8, 2, 18, 26, 8, 11, 2, 2, 32, 2, 2, 22, 8, 31, 18, 5, 32, 2, 8, 27, 24, 17, 16, 8, 32, 43, 2, 2, 44, 45, 26, 2, 8, 56, 40, 47, 32, 33, 2, 8, 64, 57, 2, 5, 36, 62, 22, 60, 8, 1, 68, 2, 56, 57, 44, 8, 32, 80, 2, 2, 8, 2, 70

Offset: 2

N. J. A. Sloane, following a suggestion of J. H. Conway, Apr 02 2004

			2^3^4^5 = 2^3^1024. But 3 == -1 (mod 4), so 3^1024 == 1 (mod 4), so 2^3^1024 == 2^1 (mod 5) since 2^4 == 1 (mod 5). Thus a(5) = 2.

Max Alekseyev, Table of n, a(n) for n = 2..1000
R. Munafo, Smallest positive integer m such that 2^3^4^5^...^n == m mod n

a(n) = n if n is a power of 2; otherwise a(n) = (2^3^4^5^...^n) mod n = A213013(n). [From Max Alekseyev, Jun 02 2012]

More terms from Robert Munafo, Apr 11 2004

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

cp's OEIS Frontend

A092188 a(n) = smallest positive integer m such that 2^3^4^5^...^n == m (mod n).

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

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