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A373206 Numbers m such that m^m == m (mod 10^(len(m) + 2)), where len(m) is the number of digits of m (A055642).

%I A373206 #33 Aug 01 2025 01:45:43
%S A373206 1,751,1001,2001,2751,3001,4001,5001,5376,6001,6751,7001,8001,9001,
%T A373206 10001,18751,20001,30001,40001,50001,58751,60001,69376,70001,80001,
%U A373206 90001,98751,100001,110001,120001,130001,138751,140001,150001,160001,170001,180001,190001
%N A373206 Numbers m such that m^m == m (mod 10^(len(m) + 2)), where len(m) is the number of digits of m (A055642).
%C A373206 By definition, this sequence is a subsequence of A082576 and also a subsequence of A373205.
%C A373206 For each integer r >= 3 the sequence contains 10^r + 1.
%C A373206 All terms > 9001 end in 0001 (e.g., 10001), 0625 (e.g., 390625), 1249 (e.g., 781249), 8751 (e.g., 18751), 9376 (e.g., 69376), and possibly in 4193, 7057, or 9375.
%H A373206 Harvey P. Dale, <a href="/A373206/b373206.txt">Table of n, a(n) for n = 1..351</a> (All terms up to 20 million)
%H A373206 <a href="/index/Ar#automorphic">Index entries for sequences related to automorphic numbers</a>
%F A373206 For all terms m, m^m == m (mod 10^(floor(log_10(m)) + 3)).
%e A373206 751 is a term since 751 is a 3-digit number and 751^751 == 500751 (mod 10^6) and thus 751^751 == 751 (mod 10^(3 + 2)).
%t A373206 Select[Range[200000],PowerMod[#,#,10^(IntegerLength[#]+2)]==#&] (* _Harvey P. Dale_, Dec 05 2024 *)
%o A373206 (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, ", "))));
%o A373206 (Python)
%o A373206 from itertools import count
%o A373206 def A373206_gen(): # generator of terms
%o A373206     yield from (1, 751, 1001, 2001, 2751, 3001, 4001, 5001, 5376, 6001, 6751, 7001, 8001, 9001)
%o A373206     for i in count(10000,10000):
%o A373206         for j in (1,625,1249,4193,7057,8751,9375,9376):
%o A373206             m = i+j
%o A373206             if pow(m,m,100*10**(len(str(m)))) == m:
%o A373206                 yield m
%o A373206 A373206_list = list(islice(A373206_gen(),20)) # _Chai Wah Wu_, Jun 02 2024
%Y A373206 Cf. A055642, A082576, A373205.
%K A373206 nonn,base
%O A373206 1,2
%A A373206 _Marco Ripà_, May 28 2024