A120962 -id:A120962 - cp's OEIS frontend

A349425 n-th stable digit (in decimal system) of n^(n^(...^n)).

1, 3, 3, 8, 0, 2, 5, 9, 3, 0, 7, 4, 5, 8, 6, 7, 8, 3, 1, 0, 1, 4, 4, 8, 7, 8, 0, 6, 4, 0, 9, 8, 0, 1, 0, 3, 5, 8, 7, 0, 6, 8, 2, 4, 1, 0, 3, 3, 3, 0, 2, 3, 5, 2, 3, 6, 8, 7, 0, 0, 7, 3, 3, 5, 0, 8, 0, 0, 8, 0, 7, 8, 8, 0, 3, 0, 6, 9, 1, 0, 7, 7, 6, 6, 5, 7, 3

Offset: 1

Marco Ripà, Nov 17 2021

The integer tetration (or hyper-4) n^^b is characterized by a well-known property involving its rightmost digits (as b grows an increasing number of the rightmost digits of n^^b are frozen - following the general rule described by Equation (16) of the linked paper "Number of stable digits of any integer tetration", p. 454).
In 2011 Ripà conjectured that, for any n >= 1, if b >= n + 2, then the n rightmost digits of n^^b are stable.
The above-mentioned paper, published in 2022, proved that this conjecture is true and also stated the stronger sufficient condition that the height of the hyperexponent is greater than or equal to tilde(v(a)) + 2, where tilde(v(a)) := v_5(a - 1) iff a == 1 (mod 5), v_5(a^2 + 1) iff a == {2, 3} (mod 5), v_5(a + 1) iff a == 4 (mod 5), v_2(a^2 - 1) - 1 iff a == 5 (mod 10), where v_2(x) = A007814(x) and v_5(x) = A112765(x) are the 2-adic and 5-adic valuations of x, respectively. - Marco Ripà, Jul 24 2024

			For n = 3, a(3) = 3 since 3^^5 == 387(mod 10^3). Thus, (387(mod 10^3) - 387(mod 10^2))/10^2 = 3.

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

Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, 2020, 26(3), 245-260.
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. A007814, A112765, A120962, A317824, A317903, A317905, A369624, A371048, A371078, A373387.

b:= proc(n) option remember; local m, v, w; m, w:= 10^n, n;
      do v:= n&^w mod m; if w=v then return v else w:=v fi od
    end:
a:= n-> `if`(irem(n, 10)=0, 0, iquo(b(n), 10^(n-1))):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 17 2021

def A349425(n):
    if n % 10 == 0: return 0
    m, n1, n2 = n, 10**n, 10**(n-1)
    while (k := pow(n,m,n1)) != m: m = k
    return k//n2 # Chai Wah Wu, Dec 19 2021

a(n) = (n^^(n + 2)(mod 10^n) - n^^(n + 2)(mod 10^(n - 1)))/10^(n - 1).

A229522 Final digit (in decimal system) of (n^n)^n, i.e., (n^n)^n mod 10.

Original entry on oeis.org

1, 6, 3, 6, 5, 6, 7, 6, 9, 0, 1, 6, 3, 6, 5, 6, 7, 6, 9, 0, 1, 6, 3, 6, 5, 6, 7, 6, 9, 0, 1, 6, 3, 6, 5, 6, 7, 6, 9, 0, 1, 6, 3, 6, 5, 6, 7, 6, 9, 0, 1, 6, 3, 6, 5, 6, 7, 6, 9, 0, 1, 6, 3, 6, 5, 6, 7, 6, 9, 0, 1, 6, 3, 6, 5, 6, 7, 6, 9, 0, 1, 6, 3, 6, 5, 6

Offset: 1

José María Grau Ribas, Sep 25 2013

Periodic sequence with period 10.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A002489, A120962, A364789 (initial digit).

Mathematica
```
Table[PowerMod[n^n, n, 10], {n, 200}]
```

a(n)=n%=10; lift(Mod(n,10)^n^n) \\ Charles R Greathouse IV, Dec 27 2013

def A229522(n): return (0, 1, 6, 3, 6, 5, 6, 7, 6, 9)[n%10] # Chai Wah Wu, Aug 10 2023

a(n) = A120962(A002489(n)). - Michel Marcus, Aug 09 2023

A229784 a(n) = (1^1^1 + 2^2^2 . . . + n^n^n) mod 10.

Original entry on oeis.org

0, 1, 7, 4, 0, 5, 1, 4, 0, 9, 9, 0, 6, 9, 5, 0, 6, 3, 9, 8, 8, 9, 5, 2, 8, 3, 9, 2, 8, 7, 7, 8, 4, 7, 3, 8, 4, 1, 7, 6, 6, 7, 3, 0, 6, 1, 7, 0, 6, 5, 5, 6, 2, 5, 1, 6, 2, 9, 5, 4, 4, 5, 1, 8, 4, 9, 5, 8, 4, 3, 3, 4, 0, 3, 9, 4, 0, 7, 3, 2, 2, 3, 9, 6, 2, 7, 3, 6, 2, 1, 1, 2, 8, 1, 7, 2, 8, 5, 1, 0, 0, 1, 7, 4, 0, 5, 1, 4, 0, 9, 9

Offset: 0

José María Grau Ribas, Sep 29 2013

The last digit of 1^1^1 + 2^2^2 +...+ n^n^n, which has period 100.

Sum of A120962 mod 10. - T. D. Noe, Sep 30 2013

Ray Chandler, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1).

Cf. A120962, A185353.

Table[Mod[Sum[PowerMod[i, i^i, 10], {i, 1, n}], 10], {n, 200}]
Mod[Accumulate[Table[PowerMod[i, i^i, 10], {i, 100}]], 10] (* T. D. Noe, Sep 30 2013 *)

a(n)=lift(sum(k=1,n%100,Mod(k,10)^k^k)) \\ Charles R Greathouse IV, Dec 27 2013

from itertools import count, accumulate, islice
def A229784_gen(): # generator of terms
    yield from accumulate((pow(k,k**k,10) for k in count(1)),func=lambda x,y:(x+y)%10)
A229784_list = list(islice(A229784_gen(),40)) # Chai Wah Wu, Jun 17 2022

a(0)=0 prepended by Alois P. Heinz, Jun 17 2022

A365689 Final decimal digit of n^((n+1)^(n+2)) = A030198(n).

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 1, 4, 5, 6

Offset: 0

Marco Ripà, Sep 16 2023

Period 10, repeat: [0, 1, 2, 1, 4, 5, 6, 1, 8, 1].

			For n = 2, a(2) = 2417851639229258349412352 mod 10 = 2.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A030198, A103562, A120962, A365689 (initial digit).

PadRight[{},100,{0,1,2,1,4,5,6,1,8,1}] (* Paolo Xausa, Oct 16 2023 *)

a(n) = lift(Mod(n, 10)^((n+1)^(n+2))); \\ Michel Marcus, Sep 16 2023

def A365689(n): return pow(n,(n+1)**(n+2),10) # Chai Wah Wu, Sep 22 2023

a(n) = n^((n+1)^(n+2)) mod 10.
a(n) = A103562(n) for n >= 4 (as 3^(2^1) == 9 (mod 10) instead of 1).
G.f.: x*(x^8+8*x^7+x^6+6*x^5+5*x^4+4*x^3+x^2+2*x+1)/(1-x^10). - Alois P. Heinz, Apr 18 2025

A365935 Final digit (in decimal system) of n^(n+1) = A007778(n).

Original entry on oeis.org

0, 1, 8, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 9, 4, 5, 6, 9, 2, 1, 0, 1, 8, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 9, 4, 5, 6, 9, 2, 1, 0, 1, 8, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 9, 4, 5, 6, 9, 2, 1, 0, 1, 8, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 9, 4, 5, 6, 9, 2, 1, 0, 1, 8, 1, 4, 5, 6

Offset: 0

Marco Ripà, Sep 23 2023

This is a periodic sequence (1, 8, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 9, 4, 5, 6, 9, 2, 1, 0) with period 20 (which is twice the base).

			For n = 3, a(3) = 3^4 mod 10 = 81 mod 10 = 1.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A007778, A010879, A056849, A120962, A365707.

a[n_]:=Last[IntegerDigits[n^(n+1)]]; Array[a,87,0] (* Stefano Spezia, Sep 26 2023 *)

a(n) = lift(Mod(n, 10)^(n+1)); \\ Michel Marcus, Sep 23 2023

a(n) = n^(n+1) mod 10.
a(n) = A010879(A007778(n)).
a(n) = A365936(n+10).

A365936 Final digit (in decimal system) of n^(n-1) = A000169(n).

Original entry on oeis.org

1, 2, 9, 4, 5, 6, 9, 2, 1, 0, 1, 8, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 9, 4, 5, 6, 9, 2, 1, 0, 1, 8, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 9, 4, 5, 6, 9, 2, 1, 0, 1, 8, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 9, 4, 5, 6, 9, 2, 1, 0, 1, 8, 1, 4, 5, 6, 1, 8, 1, 0, 1, 2, 9, 4, 5, 6, 9

Offset: 1

Marco Ripà, Sep 23 2023

This is a periodic sequence with period 20 which is twice the considered radix.

			For n = 4, a(4) = 4^3 mod 10 = 64 mod 10 = 4.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000169, A056849, A120962, A138029, A365707.

a[n_]:=Last[IntegerDigits[n^(n-1)]]; Array[a,87] (* Stefano Spezia, Sep 26 2023 *)

a(n) = n^(n-1) mod 10.

a(n) = A365935(n+10).

cp's OEIS Frontend

A349425 n-th stable digit (in decimal system) of n^(n^(...^n)).

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A229522 Final digit (in decimal system) of (n^n)^n, i.e., (n^n)^n mod 10.

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A229784 a(n) = (1^1^1 + 2^2^2 . . . + n^n^n) mod 10.

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A365689 Final decimal digit of n^((n+1)^(n+2)) = A030198(n).

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A365935 Final digit (in decimal system) of n^(n+1) = A007778(n).

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A365936 Final digit (in decimal system) of n^(n-1) = A000169(n).

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