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

1,2

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)

References

Suggested by Herb Conn.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Index entries for sequences related to automorphic numbers

Formula

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

Example

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A002283.

Sequence in context: A220968 A179662 A137859 * A177731 A074185 A242733

Adjacent sequences: A082573 A082574 A082575 * A082577 A082578 A082579

Keyword

nonn,base

Author

Gary W. Adamson, May 07 2003

Extensions

More terms from David Wasserman, Oct 27 2006

Status

approved