A322489 -id:A322489 - cp's OEIS frontend

A116081 Final nonzero digit of n^n.

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

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

cp's OEIS Frontend

A116081 Final nonzero digit of n^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Julia

Magma

Maple

Mathematica

Maxima

PARI

PARI

Python

Sage

Formula