A158624 Upper limit of backward value of 5^n.

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

Offset: 0

Author

Simon Plouffe, Mar 23 2009

Keywords

• cons
• nonn
• base
• nice

Comments

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.

Examples

			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...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..999

Crossrefs

Cf. A158625, A071583, A145679.

Programs

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

Maple

A158624:= proc(N)
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);
end proc; # Robert Israel, Apr 01 2012

Mathematica

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 *)