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

1,2

Comments

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

Links

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

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Prog

(PARI) f(n) = while(!(n % 10), n/=10); n % 10; \\ A065881
a(n) = lift(Mod(f(n), 10)^n); \\ Michel Marcus, Sep 13 2022
(PARI) a(n)=my(k=n/10^valuation(n, 10)); lift(Mod(k, 10)^(n%4+4)) \\ Charles R Greathouse IV, Sep 13 2022
(Python)
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
(Python)
def A116081(n): return pow(int(str(n).rstrip('0')[-1]), n, 10) # Chai Wah Wu, Dec 07 2023

Crossrefs

Cf. A008904, A010879, A065881.

Cf. A056849.

Cf. A204815, A204816, A204817, A204818, A204819, A230024, A204697.

Cf. A322489, A322490.

Sequence in context: A354828 A308366 A056849 * A361351 A275162 A105228

Adjacent sequences: A116078 A116079 A116080 * A116082 A116083 A116084

Keyword

easy,base,nonn,cons

Author

Greg Dresden, Mar 12 2006

Extensions

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

Status

approved