A082576 Numbers k such that k^k has final digits the same as all the digits of k.

1, 5, 6, 9, 11, 16, 21, 25, 31, 36, 41, 49, 51, 56, 57, 61, 71, 75, 76, 81, 91, 93, 96, 99, 101, 125, 151, 176, 193, 201, 249, 251, 301, 351, 375, 376, 401, 451, 499, 501, 551, 557, 576, 601, 625, 651, 693, 701, 749, 751, 776, 801, 851, 875, 901, 951, 976, 999

Offset: 1

Author

Gary W. Adamson, May 07 2003

Keywords

• nonn
• base

Comments

k^k^k also has the same final digits as k. - Ed Pegg Jr, Jun 27 2013

For any positive integer r the sequence contains 10^r-1. - Reiner Moewald, Feb 14 2016

From Robert Israel, Mar 04 2016: (Start)

All terms > 96 end in 01, 25, 49, 51, 57, 75, 76, 93 or 99.

It appears that except for 1, 5, 6, 9, 57 and 93, if k is a term then so is the number obtained from k by deleting its first digit. (End)

The only known composite and prime numbers such that n^n ends in twice the string of digits of n are 16 and 499, respectively [Gupta, 2025]. (Checked up to 10^9.) - M. F. Hasler, Jun 03 2025

Examples

			9^9 = 387420489 ends in 9, so 9 is a term.
11^11 = 285311670611 ends in 11, so 11 is a term.
		

References

• Suggested by Herb Conn.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Shyam Sunder Gupta, Do You Know, entry for 24 March 2025.
• Index entries for sequences related to automorphic numbers

Crossrefs

Cf. A002283.

Programs

Maple

a:= proc(n) option remember; local k; for k from 1+
      a(n-1) while k&^k mod (10^length(k))<>k do od; k
    end: a(1):=1:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 27 2013
select(n -> n&^n mod 10^(1+ilog10(n)) = n, [$1..1000]); # Robert Israel, Mar 04 2016

Mathematica

Select[Range@ 1000, Function[k, Take[IntegerDigits[#^#], -Length@ k] == k]@ IntegerDigits@ # &] (* Michael De Vlieger, Mar 04 2016 *)
Select[Range[1000],PowerMod[#,#,10^IntegerLength[#]]==#&] (* Harvey P. Dale, Dec 21 2019 *)

PARI

for (d = 1, 4, for (i = 10^(d - 1), 10^d - 1, x = Mod(i, 10^d); if (x^i == x, print(i)))) \\ David Wasserman, Oct 27 2006

PARI

is(n)=my(d=digits(n));Mod(n,10^#d)^n==n \\ Charles R Greathouse IV, Jan 02 2013

PARI

select( {is_A082576(n)=Mod(n,10^logint(10*n,10))^n==n}, [1..999]) \\ M. F. Hasler, Jun 03 2025

Python

from itertools import count
def A082576_gen(): # generator of terms
    yield from (1, 5, 6, 9, 11, 16, 21, 25, 31, 36, 41, 49, 51, 56, 57, 61, 71, 75, 76, 81, 91, 93, 96, 99)
    for i in count(100,100):
        for j in (1, 25, 49, 51, 57, 75, 76, 93, 99):
            m = i+j
            if pow(m,m,10**(len(str(m)))) == m:
                yield m
A082576_list = list(islice(A082576_gen(),50)) # Chai Wah Wu, Jun 02 2024

Formula

{ k : k^k mod 10^(1+floor(log_10(k))) = k }. - Jon E. Schoenfield, Jun 02 2024

Extensions

More terms from David Wasserman, Oct 27 2006