A056849 -id:A056849 - cp's OEIS frontend

A061505 Leading digit of n^n.

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

Offset: 0

Amarnath Murthy, May 06 2001

The sequence equals 2 in the range n = 3628..3730. - Rémy Sigrist, Dec 13 2018

			a(7) = 8, as 7^7 = 823543.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A056849.

a = {}; Do[ a = Append[ a, IntegerDigits[ n^n ] [ [ 1 ] ] ], {n, 1, 75 } ]; a

a(n) = digits(n^n)[1] \\ Rémy Sigrist, Dec 13 2018

Using the formula in A000030: a(n) = [n^n / 10^([log_10(n^n)])] = [n^n / 10^([n*log_10(n)])].

More terms from Robert G. Wilson v, May 10 2001
Further terms from Asher Auel, May 20 2001
a(0) = 1 prepended by Rémy Sigrist, Dec 13 2018

A116081 Final nonzero digit of n^n.

Original entry on oeis.org

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

Offset: 1

Greg Dresden, Mar 12 2006

The decimal number .147656369116... formed from these digits is a transcendental number; see Dresden's second article. These digits are never eventually periodic.

Digits appear with predictable frequencies: 1/10 for 3, 4, and 7; 1/9 for 5; 3/25 for 9; 28/225 for 1; and 307/900 for 6. - Charles R Greathouse IV, Oct 03 2022

			a(4) = 6 because 4^4 (which is 256) ends in 6.

T. D. Noe, Table of n, a(n) for n = 1..1000
Articles can be found on Dresden's Home Page.
Gregory P. Dresden, Two Irrational Numbers From the Last Non-Zero Digits of n! and n^n, Math. Mag. 74 (October 2001), 316-320.
Gregory P. Dresden, Three transcendental numbers from the last non-zero digits of n^n, F_n and n!, Mathematics Magazine, pp. 96-105, vol. 81, 2008.
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.
S. Ikeda and K. Matsuoka, On transcendental numbers generated by certain integer sequences, Siauliai Math. Semin. 8 (2013), 63-69. Mentions this sequence.
Index entries for transcendental numbers

Cf. A008904, A010879, A065881.
Cf. A056849.
Cf. A204815, A204816, A204817, A204818, A204819, A230024, A204697.
Cf. A322489, A322490.

f:= proc(n) local d, m, p; d:= min(padic:-ordp(n,2), padic:-ordp(n,5));
   m:= n/10^d;
   p:= n - 1 mod 4 + 1;
   m &^ p mod 10;
end proc:
seq(f(n), n=1..1000); # Robert Israel, Oct 19 2014

f[n_] := Block[{m = n}, While[ Mod[m, 10] == 0, m /= 10]; PowerMod[m, n, 10]]; Array[f, 105] (* Robert G. Wilson v, Mar 13 2006 and modified Oct 12 2014 *)

f(n) = while(!(n % 10), n/=10); n % 10; \\ A065881
a(n) = lift(Mod(f(n), 10)^n); \\ Michel Marcus, Sep 13 2022

a(n)=my(k=n/10^valuation(n,10)); lift(Mod(k,10)^(n%4+4)) \\ Charles R Greathouse IV, Sep 13 2022

def a(n):
    k = n
    while k%10 == 0: k //= 10
    return pow(k, n, 10)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Sep 13 2022

def A116081(n): return pow(int(str(n).rstrip('0')[-1]),n,10) # Chai Wah Wu, Dec 07 2023

a(n) = A065881(n)^n mod 10 = A010879(A065881(n)^(A010883(n-1))). - Robert Israel, Oct 19 2014

More terms from Robert G. Wilson v, Mar 13 2006

A322489 Numbers k such that k^k ends with 4.

Original entry on oeis.org

2, 18, 22, 38, 42, 58, 62, 78, 82, 98, 102, 118, 122, 138, 142, 158, 162, 178, 182, 198, 202, 218, 222, 238, 242, 258, 262, 278, 282, 298, 302, 318, 322, 338, 342, 358, 362, 378, 382, 398, 402, 418, 422, 438, 442, 458, 462, 478, 482, 498, 502, 518, 522, 538, 542, 558

Offset: 1

Bruno Berselli, Dec 12 2018

Also numbers k == 2 (mod 4) such that 2^k and k^2 end with the same digit.

Numbers congruent to {2, 18} mod 20. - Amiram Eldar, Feb 27 2023

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004526, A056849.
Subsequence of A139544, A235700.
Numbers k such that k^k ends with d: A008592 (d=0), A017281 (d=1), A067870 (d=3), this sequence (d=4), A017329 (d=5), A271346 (d=6), A322490 (d=7), A017377 (d=9).

GAP
```
List([1..70], n -> 10*n+3*(-1)^n-5);
```

[10*n+3*(-1)^n-5 for n in 1:70] |> println

Magma
```
[10*n+3*(-1)^n-5: n in [1..70]];
```

select(n->n^n mod 10=4,[$1..558]); # Paolo P. Lava, Dec 18 2018

Mathematica
```
Table[10 n + 3 (-1)^n - 5, {n, 1, 60}]
```
Maxima
```
makelist(10*n+3*(-1)^n-5, n, 1, 70);
```

apply(A322489(n)=10*n+3*(-1)^n-5, [1..70]) \\ M. F. Hasler, Dec 14 2018

Vec(2*x*(1 + 8*x + x^2) / ((1 - x)^2*(1 + x)) + O(x^70)) \\ Colin Barker, Dec 13 2018

[10*n+3*(-1)**n-5 for n in range(1, 70)]

Sage
```
[10*n+3*(-1)^n-5 for n in (1..70)]
```

O.g.f.: 2*x*(1 + 8*x + x^2)/((1 + x)*(1 - x)^2).
E.g.f.: 2 + 3*exp(-x) + 5*(2*x - 1)*exp(x).
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 10*n + 3*(-1)^n - 5. Therefore:
a(n) = 10*n - 8 for odd n;
a(n) = 10*n - 2 for even n.
a(n+2*k) = a(n) + 20*k.
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(2*Pi/5)*Pi/20 = sqrt(5+2*sqrt(5))*Pi/20. - Amiram Eldar, Feb 27 2023

A322490 Numbers k such that k^k ends with 7.

Original entry on oeis.org

3, 17, 23, 37, 43, 57, 63, 77, 83, 97, 103, 117, 123, 137, 143, 157, 163, 177, 183, 197, 203, 217, 223, 237, 243, 257, 263, 277, 283, 297, 303, 317, 323, 337, 343, 357, 363, 377, 383, 397, 403, 417, 423, 437, 443, 457, 463, 477, 483, 497, 503, 517, 523, 537, 543, 557, 563

Offset: 1

Bruno Berselli, Dec 12 2018

Equivalently, numbers k such that k and (7^h)^k end with the same digit, where h == 1 (mod 4).
Also, numbers k such that k and (3^h)^k end with the same digit, where h == 3 (mod 4).
Numbers congruent to {3, 17} mod 20. - Amiram Eldar, Feb 27 2023

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004526, A056849.
Subsequence of A063226, A295009.
Similar sequences are listed in A322489.

GAP
```
List([1..70], n -> 10*n+2*(-1)^n-5);
```

[10*n+2*(-1)^n-5 for n in 1:70] |> println

Magma
```
[10*n+2*(-1)^n-5: n in [1..70]];
```

select(n->n^n mod 10=7,[$1..563]); # Paolo P. Lava, Dec 18 2018

Table[10 n + 2 (-1)^n - 5, {n, 1, 60}]
LinearRecurrence[{1,1,-1},{3,17,23},80] (* Harvey P. Dale, Sep 15 2019 *)

Maxima
```
makelist(10*n+2*(-1)^n-5, n, 1, 70);
```

apply(A322490(n)=10*n+2*(-1)^n-5, [1..70])

Vec(x*(3 + 14*x + 3*x^2) / ((1 + x)*(1 - x)^2) + O(x^55)) \\ Colin Barker, Dec 13 2018

[10*n+2*(-1)**n-5 for n in range(1, 70)]

Sage
```
[10*n+2*(-1)^n-5 for n in (1..70)]
```

O.g.f.: x*(3 + 14*x + 3*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: 3 + 2*exp(-x) + 5*(2*x - 1)*exp(x).
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 10*n + 2*(-1)^n - 5. Therefore:
a(n) = 10*n - 7 for odd n;
a(n) = 10*n - 3 for even n.
a(n+2*k) = a(n) + 20*k.
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(7*Pi/20)*Pi/20. - Amiram Eldar, Feb 27 2023

A238291 Final digit of real part of (n+n*i)^n.

Original entry on oeis.org

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

Offset: 1

José María Grau Ribas, Feb 22 2014

Starting from a(2) the sequence is periodic with period length 20. - Giovanni Resta, Feb 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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. A238292, A056849.

re[n_, b_] := Mod[b + Re@PowerMod[(n + n I), n, b], b];
Table[re[n, 10], {n, 100}]

A238292 Final digit of imaginary part of (n+n*i)^n.

Original entry on oeis.org

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

Offset: 1

José María Grau Ribas, Feb 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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. A238291, A056849.

im[n_, b_] := Mod[b + Im@PowerMod[(n + n I), n, b], b]; Table[im[n, 10], {n, 100}]

A271346 Numbers k such that the final digit of k^k is 6.

Original entry on oeis.org

4, 6, 8, 12, 14, 16, 24, 26, 28, 32, 34, 36, 44, 46, 48, 52, 54, 56, 64, 66, 68, 72, 74, 76, 84, 86, 88, 92, 94, 96, 104, 106, 108, 112, 114, 116, 124, 126, 128, 132, 134, 136, 144, 146, 148, 152, 154, 156, 164, 166, 168, 172, 174, 176, 184, 186, 188, 192, 194

Offset: 1

Wesley Ivan Hurt, Apr 04 2016

The values of n^n (A000312) end in every digit except for 2 and 8. The sequence of final digits of n^n (A056849) is periodic with period 20; for n=1,2,... the last digits are [1, 4, 7, 6, 5, 6, 3, 6, 9, 0, 1, 6, 3, 6, 5, 6, 7, 4, 9, 0]. Thus, 6 is the most common final digit of n^n. Since 6 does not occur at any odd index in the list above, all terms of a(n) are even. Also, from the distribution of 6's in the list, we can see that the difference between any two consecutive values of a(n) will be 2, 4 or 8.

Felix Fröhlich, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A000312 (n^n), A056849 (final digit of n^n).

I:=[4,6,8,12,14,16,24]; [n le 7 select I[n] else Self(n-1)+Self(n-6)-Self(n-7): n in [1..60]]; // Vincenzo Librandi, Oct 09 2017

A271346:=n->`if`(n^n mod 10 = 6, n, NULL): seq(A271346(n), n=1..500);

LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {4, 6, 8, 12, 14, 16, 24},59] (* Ray Chandler, Mar 08 2017 *)

is(n) = Mod(n, 10)^n==6 \\ Felix Fröhlich, Apr 07 2016

Vec(2*x*(1 + x^2)*(2 + x - x^2 + x^3 + 2*x^4) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^59)) \\ Colin Barker, Dec 13 2018

a(n) = a(n-1) + a(n-6) - a(n-7) for n > 7. - Wesley Ivan Hurt, Oct 08 2017

G.f.: 2*x*(1 + x^2)*(2 + x - x^2 + x^3 + 2*x^4) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)). - Colin Barker, Dec 13 2018

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).

A175348 Last digit of p^p, where p is the n-th prime.

Original entry on oeis.org

4, 7, 5, 3, 1, 3, 7, 9, 7, 9, 1, 7, 1, 7, 3, 3, 9, 1, 3, 1, 3, 9, 7, 9, 7, 1, 7, 3, 9, 3, 3, 1, 7, 9, 9, 1, 7, 7, 3, 3, 9, 1, 1, 3, 7, 9, 1, 7, 3, 9, 3, 9, 1, 1, 7, 7, 9, 1, 7, 1, 7, 3, 3, 1, 3, 7, 1, 7, 3, 9, 3, 9, 3, 3, 9, 7, 9, 7, 1, 9, 9, 1, 1, 3, 9, 7, 9, 7, 1, 7, 3, 9, 3, 1, 9, 7, 9, 1, 7, 1, 3, 7, 7, 9, 1

Offset: 1

Charles R Greathouse IV, Apr 19 2010

Euler and Sadek ask whether the sequence, interpreted as the decimal expansion N = 0.47531..., is rational or irrational.

Dickson's conjecture implies that each finite sequence with values in {1,3,7,9} occurs as a substring. In particular, this implies that the above N is irrational. - Robert Israel, Jan 26 2017

			prime(4) = 7 and 7^7 = 823543, so a(4) = 3.

R. Euler and J. Sadek, A number that gives the unit digit of n^n. Journal of Recreational Mathematics, 29:3 (1998), pp. 203-204.

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

Cf. A000040, A007652, A056849, A137807.

R:= [seq(i &^ i mod 10, i=1..20)]:
seq(R[ithprime(i) mod 20],i=1..100); # Robert Israel, Jan 26 2017

Table[PowerMod[n,n,10],{n,Prime[Range[110]]}] (* Harvey P. Dale, Mar 24 2024 *)

a(n)=[1,4,7,0,5,0,3,0,9,0,1,0,3,0,0,0,7,0,9][prime(n)%20]

a(n) = A056849(A000040(n)). - Robert Israel, Jan 26 2017

A229481 Final digit of 1^n + 2^n + ... + n^n.

Original entry on oeis.org

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

Offset: 0

José María Grau Ribas, Sep 24 2013

Cyclic with period 100.

Alois P. Heinz, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A031971, A056849.

a:= proc(n) local l; l:=[seq(add(k&^i mod 10, k=1..i) mod 10, i=0..99)]:
      proc(n) l[1+irem(n, 100)] end
    end():
seq(a(n), n=0..200);  # Alois P. Heinz, Sep 26 2013

Table[Mod[Sum[PowerMod[i, n, 10], {i, 1, n}], 10], {n, 0, 133}]

a(n)=n%=100;lift(sum(k=1,n,Mod(k,10)^n)) \\ Charles R Greathouse IV, Dec 13 2013

a(n) = a(n-100). - Wesley Ivan Hurt, Jan 02 2024

cp's OEIS Frontend

A061505 Leading digit of n^n.

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A116081 Final nonzero digit of n^n.

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A322489 Numbers k such that k^k ends with 4.

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A322490 Numbers k such that k^k ends with 7.

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GAP

Julia

Magma

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Maxima

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PARI

Python

Sage

Formula

A238291 Final digit of real part of (n+n*i)^n.

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Mathematica

A238292 Final digit of imaginary part of (n+n*i)^n.

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