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

Simon Plouffe simon.plouffe at gmail.com
Mon Mar 23 07:13:16 CET 2009

```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.

#################################################################################################

These are additional comments and values to sequence A023415

%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.

```