A373205 Numbers m such that m^m == m (mod 10^(len(m) + 1)), where len(m) is the number of digits of m (A055642).

1, 51, 57, 101, 151, 176, 201, 301, 351, 401, 501, 551, 576, 601, 625, 701, 751, 801, 901, 951, 976, 1001, 1376, 2001, 2057, 2751, 3001, 4001, 4193, 4751, 5001, 5376, 6001, 6249, 6751, 7001, 8001, 8751, 9001, 9375, 9376, 10001, 10751, 11001, 12001, 13001

Offset: 1

Author

Marco Ripà, May 27 2024

Keywords

• nonn
• base

Comments

By definition, the present sequence is a subsequence of A082576.

For each integer r >= 2 this sequence contains 10^r + 1.

All terms > 1 end in 01, 25, 49, 51, 57, 75, 76, or 93.

Examples

			51 is a term since 51 is a 2-digit number and 51^51 == 5051 (mod 10^4) and thus 51^51 == 51 (mod 10^(2 + 1)).
		

Links

• Index entries for sequences related to automorphic numbers

Crossrefs

Cf. A055642, A082576, A373206.

Programs

PARI

for (len_m = 1, 5, for (m = 10^(len_m - 1), 10^len_m - 1, if (m == Mod(m, 10^(len_m + 1))^m, print1(m, ", "))))

Python

from itertools import count
def A373205_gen(): # generator of terms
    for i in count(0,100):
        for j in (1, 25, 49, 51, 57, 75, 76, 93):
            m = i+j
            if pow(m,m,10*10**(len(str(m)))) == m:
                yield m
A373205_list = list(islice(A373205_gen(),20)) # Chai Wah Wu, Jun 02 2024