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A158625 Lower limit of backward value of 5^n.

%I A158625 #82 Aug 02 2024 12:29:00
%S A158625 5,2,1,3,0,2,3,3,0,4,3,1,1,3,1,1,2,4,2,1,0,3,1,3,3,0,0,0,2,3,1,4,1,0,
%T A158625 2,1,0,3,4,3,0,2,1,2,2,1,1,4,4,3,4,0,2,0,4,0,2,2,1,1,1,1,1,0,1,0,0,1,
%U A158625 1,3,3,2,0,1,1,4,4,2,0,1,4,2,4,1,2,0,4
%N A158625 Lower limit of backward value of 5^n.
%C A158625 Digits are all in {0,1,2,3,4} after the first term.
%C A158625 The upper limit is A158624, 0.5265679578796997657885576975995789586775656...
%C A158625 The sequence is not eventually periodic. Assuming any period results in a condition a(1)=0 mod 10 which contradicts a(1)=5. - _Cezary Glowacz_, Jul 22 2024
%H A158625 Jon E. Schoenfield, <a href="/A158625/b158625.txt">Table of n, a(n) for n = 1..3000</a>
%F A158625 a(n) >= 0 and is the minimum satisfying (Sum_{i=1..n} a(i)*10^(i-1)) == 0 (mod 5^n), for n >= 2. - _Cezary Glowacz_, Jul 24 2024
%e A158625 5^3 = 125 so the backward value is 0.521, 5^10 = 9765625, so backward value is 0.5265679. The lower limit of all values is a constant, which appears to be 0.521302330431131124210313300023141021034302...
%e A158625 From _N. J. A. Sloane_, May 11 2018: (Start)
%e A158625 To describe this another way:  write down the decimal expansion of powers of 5:
%e A158625   1
%e A158625   5
%e A158625   25
%e A158625   125
%e A158625   625
%e A158625   3125
%e A158625   ...
%e A158625 keep going forever.
%e A158625 Write them backwards:
%e A158625   1
%e A158625   5
%e A158625   52
%e A158625   526
%e A158625   5213
%e A158625   ...
%e A158625 After a while the beginning digits are all the same.
%e A158625 That's the sequence. (End)
%o A158625 (Python)
%o A158625 # lower limit of backward sequence of 5^n
%o A158625 a,i=5,0; x=a
%o A158625 while i < 100:
%o A158625      i+=1; print(x, end=',')
%o A158625      x=(-a//pow(5,i)*pow(3,i))%5; a+=x*pow(10,i)
%o A158625 # _Cezary Glowacz_, Jul 29 2024
%o A158625 (Magma) D:=87; e:=6; for d in [2..D-1] do t:=Modexp(5,e,10^(d+1)); if t div 10^d ge 5 then e+:=2^(d-2); end if; end for; t:=Modexp(5,e,10^D); IntegerToSequence(t,10); // _Jon E. Schoenfield_, Feb 05 2018
%Y A158625 Cf. A158624, A004094, A023415, A145679.
%K A158625 cons,nonn,nice,base
%O A158625 1,1
%A A158625 _Simon Plouffe_, Mar 23 2009