A061505 -id:A061505 - cp's OEIS frontend

A056849 Final digit of n^n.

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

Offset: 1

Robert G. Wilson v, Aug 30 2000

Cyclic with a period of 20.

Also decimal expansion of 147656369016365674900/(10^20-1). - Bruno Berselli, Sep 27 2021

R. Euler and J. Sadek, "A Number That Gives The Units Of n^n", Journal of Recreational Mathematics, vol. 29(3), 1998, pp. 203-4.

Hung Viet Chu, New Transcendental Numbers from Certain Sequences, arXiv:1908.03855 [math.NT], 2019. Mentions this sequence.
Gregory P. Dresden, Three transcendental numbers from the last non-zero digits of n^n, F_n and n!, Mathematics Magazine, vol. 81, 2008, pp. 96-105.
Gregory Dresden, Two Irrational Numbers That Give the Last Non-Zero Digits of n! and n^n, arXiv:1904.10274 [math.NT], 2019.
Jose María Grau and A. M. Oller-Marcen, On the last digit and the last non-zero digit of n^n in base b, arXiv:1203.4066 [math.NT], 2012.
Jose María Grau and A. M. Oller-Marcen, On the last digit and the last non-zero digit of n^n in base b, Bull. Korean Math. Soc. 51 (2014), No. 5, pp. 1325-1337.
Index entries for sequences related to final digits of numbers
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. A000312, A061505, A116081.

n^n mod 3..9: A204688, A204689, A204690, A204671, A204693, A204694, A204695.

[Modexp(n, n, 10): n in [1..100]]; // Bruno Berselli, Sep 27 2021

Maple
```
seq(n &^ n mod 10, n=1..120);
```
Mathematica
```
Table[PowerMod[n, n, 10], {n, 1, 100}]
```

a(n)=lift(Mod(n,10)^n) \\ Charles R Greathouse IV, Dec 29 2012

def a(n): return pow(n, n, 10)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Sep 13 2022

A120962 Final digit (in decimal system) of n^(n^n), i.e., n^(n^n) mod 10.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 19 2006, Oct 26 2007

Periodic sequence with period length 20. - Arkadiusz Wesolowski, Feb 12 2012

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

Cf. A061505, A229522, A229784.

Maple
```
seq(n &^ (n^n) mod 10, n=0..105);
```

Join[{0}, Table[PowerMod[n, n^n, 10], {n, 100}]] (* Stefan Steinerberger, Nov 23 2007 *)

a(n)=if(n%10,lift(Mod(n,10)^lift(Mod(n,20)^n)),0) \\ Charles R Greathouse IV, Feb 12 2012

def A120962(n): return pow(n,n**n,10) # Chai Wah Wu, Sep 22 2023

a(n) = A010879(A002488(n)). - Michel Marcus, Aug 04 2015

More terms from Stefan Steinerberger, Nov 23 2007

A138029 Main diagonal of A138028; the most significant digit of n^(n-1).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Feb 10 2008

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A061505, A138028.

f:= n -> floor(10^frac((n-1)*log[10](n))):
map(f, [$1..200]); # Robert Israel, Jan 15 2019

f[n_, k_] := Quotient[n^k, 10^Floor[k*Log[10, n]]]; Table[ f[ n, n - 1], {n, 105}]

Name corrected by Robert Israel, Jan 15 2019

A365707 Initial digit of n^(n+1) (A007778(n)).

Original entry on oeis.org

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

Offset: 0

Marco Ripà, Sep 16 2023

			a(3) = 8, since 3^(3+1) = 3^4 = 81.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000030, A007778, A061505, A138029, A364855, A365935.

seq(convert(n^(n+1),base,10)[-1],n=0..100); # Robert Israel, Feb 16 2024

Join[{0}, Table[Floor[n^(n+1)/10^Floor[Log10[n^(n+1)]]], {n, 86}]]

a(n) = floor((n^(n+1))/10^floor(log_10(n^(n+1)))).

a(n) = A000030(A007778(n)).

A138028 The array of the most significant digit of n^k read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Feb 10 2008

			n\k
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... .
1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, ... .
1, 3, 9, 2, 8, 2, 7, 2, 6, 1, 5, 1, 5, 1, 4, 1, 4, 1, 3, 1, 3, ... .
1, 4, 1, 6, 2, 1, 4, 1, 6, 2, 1, 4, 1, 6, 2, 1, 4, 1, 6, 2, 1, ... .
1, 5, 2, 1, 6, 3, 1, 7, 3, 1, 9, 4, 2, 1, 6, 3, 1, 7, 3, 1, 9, ... .
1, 6, 3, 2, 1, 7, 4, 2, 1, 1, 6, 3, 2, 1, 7, 4, 2, 1, 1, 6, 3, ... .
1, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1, ... .
1, 9, 8, 7, 6, 5, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, ... .
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... .
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, ... .
1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 2, 2, 3, 3, ... .
......................................................................

Cf. A008952, A060956, A138029, A061505.

f[n_, k_] := Quotient[n^k, 10^Floor[k*Log[10, n]]]; Table[f[ n - k, k], {n, 14}, {k, 0, n - 1}] // Flatten

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A056849 Final digit of n^n.

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A120962 Final digit (in decimal system) of n^(n^n), i.e., n^(n^n) mod 10.

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A138029 Main diagonal of A138028; the most significant digit of n^(n-1).

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A365707 Initial digit of n^(n+1) (A007778(n)).

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A138028 The array of the most significant digit of n^k read by antidiagonals.

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