# [seqfan] Re: 4 interesting constants and 3 new sequences.

Hugo van der Sanden human at google.com
Mon Mar 23 12:09:53 CET 2009

```Let D_{n-1} be our limiting (minimum) value for the last n-1 digits of 2^k.
Then D_{n-1} must be divisible by 2^{n-1}. D_n is then fixed to be even or
odd according to D_{n-1} / 2^(n-1) mod 2; further, either all 5 even or all
5 odd digits provide an achievable n-digit terminator for powers of 2, so
additional digits for the minimum can only be drawn from 0 or 1.

The same happens for the limiting maximal case, and symmetry demands that
the parity of new digits will be the same as for the minimal case. Hence the
fact that they line up, and sum to a simple integer.

For 5^k, the same argument applies, with selections of digits driven in this
case by D_{n-1} / 5^(n-1) mod 5.

Hugo

2009/3/23 Simon Plouffe <simon.plouffe at gmail.com>

>
> Hello everybody,
>
>  Here are 4 strange constants (and sequences).
>
> I was studying the numbers of 5^n, 2^n and n! written backward and realized
> that
> powers of 5 appear in a stange manner (as well as many others!).
>
> Consider 5^3 = 125, so the backward value is 0.521,  5^10 = 9765625, so
> backward value is
> 0.5265679, see A071583.
>
> It appears that there is an upper limit to this process, the
> constant is 0.5265679578796997657885576975995789586775656...  the lower
> limit being
> 0.521302330431131124210313300023141021034302...
>
>
> The interesting fact is the following, after the 2nd term (or second
> digit), only the
> numbers in {5,6,7,8,9} appears.
> The constant does NOT correspond to anything I know.
> The lower limit has the same property, only the numbers in {0,1,2,3,4}
> appears after
> the first term.
>
> The phenomena appears also with powers of 2.
> The upper limit for 2^n is 0.8889899898988998  the lower limit is
> 0.21101001010110010...
>
> The number 0.8889899898988998  appear in A023415 of David Wilson.
>
> again, only the numbers in {8,9} appear in the first constant and only
> digits in {0,1} in the second (after the first digit). The most surprising
> fact
> is that the sum : 8.8898998989889987 + 2.1101001010110010 = 11 EXACTLY.
> These are experimental results as you may guess.
>
> INTERESTING NOTE : if we take backward values of n! then the 2 constants
> are
> the same and the sum is still 11 exactly. Can someone explain this strange
> phenomena ?
>
> Simon Plouffe
>
>
>
> ##############################################################################################
>
> %I A158624
> %S A158624
> 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,
> %T A158624 8,6,7,7,5,6,5,6
> %N A158624 Upper limit of backward value of 5^n.
>
> %C A158624 Digits are all in {5,6,7,8,9} after 2nd term.
> %F A158624 No known formula.
> %C A158624 The lower limit is
> 0.521302330431131124210313300023141021034302...
> %e A158624 5^3 = 125 so the backward value is 0.521, 5^10 = 9765625, so
> backward value is
> 0.5265679. The upper limit of all values is a constant, constant appears to
> be
> 0.5265679578796997657885576975995789586775656...
> %K A158624 cons,nonn
> %O A158624 1,1
> %A A158624 Simon Plouffe (simon.plouffe(AT)gmail.com), Mar 23 2009.
> %R A158624 see A158625, A071583.
>
>
> ################################################################################################
>
> %S A158625
> 5,2,1,3,0,2,3,3,0,4,3,1,1,3,1,1,2,4,2,1,0,3,1,3,3,0,0,0,2,3,1,4,1,0,2,1,0,3,4,3,0,2
> %N A158625 Lower limit of backward value of 5^n.
> %C A158625 Digits are all in {0,1,2,3,4} after first term.
> %F A158625 No known formula.
> %C A158625 The upper limit is
> 0.5265679578796997657885576975995789586775656...
> %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, constant appears to
> be
> 0.521302330431131124210313300023141021034302...
> %K A158625 cons,nonn
> %O A158625 1,1
> %R A158625 see A158624, A004094.
> %A A158625 Simon Plouffe (simon.plouffe(AT)gmail.com), Mar 23 2009.
>
>
> #################################################################################################
>
>
> %N A023415 Upper limit of backward value of 2^n and n!.
> %C A023415 The sum of constants in A158626 and A158627 is conjectured to be
> 11 exactly.
> %A A023415 Simon Plouffe (simon.plouffe(AT)gmail.com), Mar 23 2009.
>
> #################################################################################################
>
> %S A158627 2,1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0
>
> %N A158627 Lower limit of backward value of 2^n and n!.
> %C A158727 The sum of constants in A158626 and A158627 is conjectured to be
> 11 exactly.
> %A A158627 Simon Plouffe (simon.plouffe(AT)gmail.com), Mar 23 2009.
>
> important comment : the value of A158627 is deduced from sequence A023415
> if conjecture about 11 is true then the sequence A158627 would be :
> [2, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1,
> 1, 0, 1, 1, 1, 1, 0,
> 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1
> , 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1,
> 1, 0, 1, 1, 0, 0, 1,
> 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1,
> 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1]:
>
> There are no apparent pattern in this sequence.
>
>
>
>
>
>
>
>
>
>
>
>
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>
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>

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