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A158624
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Upper limit of backward value of 5^n.
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5
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5, 2, 6, 5, 6, 7, 9, 5, 7, 8, 7, 9, 6, 9, 9, 7, 6, 5, 7, 8, 8, 5, 5, 7, 6, 9, 7, 5, 9, 9, 5, 7, 8, 9, 5, 8, 6, 7, 7, 5, 6, 5, 6, 9, 5, 7, 5, 6, 6, 9, 6, 7, 7, 6, 7, 6, 8, 8, 5, 8, 5, 6, 7, 5, 8, 9, 6, 6, 7, 5, 9, 5, 7, 9, 8, 6, 8, 8, 7, 9, 5, 8, 8, 5, 8, 5, 9, 5, 5, 8, 9, 7, 7, 9, 7, 7, 9, 6, 7, 6, 8, 9, 7, 6, 6
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OFFSET
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0,1
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COMMENTS
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Digits are all in {5,6,7,8,9} after 2nd term.
The other limit, related to odd n, is in A158625.
The first digit of the backward value of 5^n is always a(0)=5. The second digit is a(-1)=2 from n=2 on. The third digit is a(-2)=6 for all even n >= 4. The fourth digit is a(-3)=5 for n=6+4k, k >= 0. The fifth digit is a(-4)=6 for n=10+8k, k >= 0. The 6th digit is a(-5)=7 for n=10+16k, k >= 0. The 7th digit is a(-6)=9 for n=10+32k, k >= 0.
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LINKS
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EXAMPLE
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5^3 = 125 so the backward value is 0.521, 5^10 = 9765625, so the backward value is 0.5265679. The upper limit of all values is a constant, which appears to be 0.5265679578796997657885576975995789586775656...
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MAPLE
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local m, n, A;
m[2]:= 3;
for n from 3 to N do
A:= 5&^m[n-1] mod 10^n;
if A > 5*10^(n-1) then m[n]:= m[n-1]
else m[n]:= m[n-1]+2^(n-3)
end if
end do:
convert(5&^m[N] mod 10^(N), base, 10);
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MATHEMATICA
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A158624[k_] := Module[{m, n, a}, m[2] = 3; For[n = 3, n <= k, n++, a = PowerMod[5, m[n-1], 10^n]; If[ a > 5*10^(n-1), m[n] = m[n-1], m[n] = m[n-1] + 2^(n-3)]]; IntegerDigits[PowerMod[5, m[k], 10^k]] // Reverse]; A158624[105] (* Jean-François Alcover, Dec 21 2012, translated from Robert Israel's Maple program *)
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PROG
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(Magma) D:=87; e:=6; for d in [2..D-1] do t:=Modexp(5, e, 10^(d+1)); if t div 10^d lt 5 then e+:=2^(d-2); end if; end for; t:=Modexp(5, e, 10^D); IntegerToSequence(t, 10); // Jon E. Schoenfield, Feb 07 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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