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A068407 Automorphic numbers: numbers k such that k^5 ends with k. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 1 (mod 4).

%I A068407 #34 Mar 12 2024 02:40:03
%S A068407 0,1,2,3,4,5,6,7,8,9,24,25,32,43,49,51,57,68,75,76,93,99,125,193,249,
%T A068407 251,307,375,376,432,443,499,501,557,568,624,625,693,749,751,807,875,
%U A068407 943,999,1249,1251,1693,1875,2057,2499,2501,2943,3125,3307,3568,3749,3751
%N A068407 Automorphic numbers: numbers k such that k^5 ends with k. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 1 (mod 4).
%H A068407 Eric M. Schmidt, <a href="/A068407/b068407.txt">Table of n, a(n) for n = 1..1000</a>
%H A068407 <a href="/index/Ar#automorphic">Index entries for sequences related to automorphic numbers</a>
%e A068407 13568 is a term because 13568^5 = 459810807237016813568 ends with 13568.
%t A068407 Select[Range[0,100000],PowerMod[#,5,10^IntegerLength[#]]==#&] (* _Harvey P. Dale_, Nov 04 2011 *)
%o A068407 (Sage)
%o A068407 def automorphic(maxdigits, pow, base=10) :
%o A068407     morphs = [[0]]
%o A068407     for i in range(maxdigits):
%o A068407         T=[d*base^i+x for x in morphs[-1] for d in range(base)]
%o A068407         morphs.append([x for x in T if x^pow % base^(i+1) == x])
%o A068407     return sorted(set(sum(morphs,[])))
%o A068407 # (call with pow=5 for this sequence), _Eric M. Schmidt_, Jul 29 2013
%o A068407 (Magma) [n : n in [0..3749] | Intseq(n^5)[1..#Intseq(n)] eq Intseq(n)]; // _Arkadiusz Wesolowski_, Nov 15 2013
%Y A068407 Cf. A003226, A033819.
%K A068407 nonn,base
%O A068407 1,3
%A A068407 _Benoit Cloitre_, Mar 08 2002