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A344823 Numbers m with decimal expansion (d_1, ..., d_k) such that d_i = m ^ i mod 10 for i = 1..k.

%I A344823 #10 May 31 2021 02:11:21
%S A344823 0,1,2,3,4,5,6,7,8,9,11,55,66,111,464,555,666,919,1111,5555,6666,
%T A344823 11111,24862,39713,46464,55555,66666,79317,84268,91919,111111,555555,
%U A344823 666666,1111111,4646464,5555555,6666666,9191919,11111111,55555555,66666666,111111111
%N A344823 Numbers m with decimal expansion (d_1, ..., d_k) such that d_i = m ^ i mod 10 for i = 1..k.
%C A344823 This sequence is infinite as it contains d * A002275(k) for any d in {1, 5, 6} and k > 0.
%C A344823 Also contains terms with patterns 4(64)^k, 9(19)^k, 2(4862)^k, 3(9713)^k, 7(9317)^k, 8(4268)^k for k >= 0, where ^ denotes repeated concatenation; all terms have first and last digits the same. - _Michael S. Branicky_, May 29 2021
%H A344823 Rémy Sigrist, <a href="/A344823/b344823.txt">Table of n, a(n) for n = 1..1253</a>
%H A344823 Rémy Sigrist, <a href="/A344823/a344823.gp.txt">PARI program for A344823</a>
%e A344823 - 4^1 = 4 mod 10,
%e A344823 - 4^2 = 6 mod 10,
%e A344823 - 4^3 = 4 mod 10,
%e A344823 - so 464 belongs to the sequence.
%o A344823 (PARI) is(n) = { my (d=digits(n), m=Mod(n,10)); for (k=1, #d, if (d[k] != m^k, return (0))); return (1) }
%o A344823 (PARI) See Links section.
%o A344823 (Python)
%o A344823 def ok(m):
%o A344823   d = str(m)
%o A344823   return all(d[i-1] == str((m**i)%10) for i in range(1, len(d)+1))
%o A344823 print(list(filter(ok, range(10**6)))) # _Michael S. Branicky_, May 29 2021
%o A344823 (Python)
%o A344823 def auptod(maxdigits):
%o A344823   alst = [0]
%o A344823   for k in range(1, maxdigits+1):
%o A344823     for d1 in range(1, 10):
%o A344823       d = [(d1**i)%10 for i in range(1, k+1)]
%o A344823       if d1 == d[-1]: alst.append(int("".join(map(str, d))))
%o A344823   return alst
%o A344823 print(auptod(9)) # _Michael S. Branicky_, May 29 2021
%Y A344823 Cf. A002275, A344555, A344749, A344822.
%K A344823 nonn,base
%O A344823 1,3
%A A344823 _Rémy Sigrist_, May 29 2021