help needed with a duplicate

Sun Jun 24 00:20:49 CEST 2007

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From: "David Wilson" <davidwwilson at comcast.net>
To: <seqfan at ext.jussieu.fr>
References: <200706232003.l5NK3KDg22565812 at fry.research.att.com> <023a01c7b5e2$0be7b780$5504b4d8 at DCR74871>
Subject: Re: help needed with a duplicate
Date: Sat, 23 Jun 2007 19:57:31 -0400
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This is an example of a poorly conceived sequence, IMHO.

The idea, n that are a substring of 2^n, is mildly interesting. Statiscally, 
n will not be a substring of 2^n most of the time (I didn't do the math, but 
empirically, the density of such numbers hangs around 1/10). This means that 
most of the elements of A124692 should be -1, as we observe.

I admit an a priori aversion to sequences of the form "a(n) = f(n), or x if 
f(n) does not exist." But in this case, the choice of x = -1 is particularly 
poor, as x = 0 would have worked just as well, without introducing %VWX 
lines in the sequence.

Unless the undefined values of f are very few and far between, it seems 
infinitely better to me to define a domain sequence d for f, then define the 
target sequence to include just those values of f on its domain, e.g, 
f(d(n)).

In this case, we would define the domain sequence:

%N A000000 n is a substring of 2^n
%S A000000 
6,10,35,36,37,44,49,51,60,67,72,73,82,85,89,93,179,188,190,191,226,234,

and then give f in terms of its domain:

%N A124692 n is a substring of of 2^A000000(n) starting at the a(n)th digit 
from the left.
%S A124692 
1,1,3,10,2,10,5,3,11,13,2,5,20,4,3,24,19,5,17,19,35,25,53,67,60,61,33,74,18,109,

----- Original Message ----- 
From: "Jon Schoenfield" <jonscho at hiwaay.net>
To: <njas at research.att.com>; <seqfan at ext.jussieu.fr>
Cc: <hillcino368 at hotmail.com>
Sent: Saturday, June 23, 2007 6:01 PM
Subject: Re: help needed with a duplicate

> Neil,
>
> <<  it seems to me a(11) = -1 since 2^11 = 2048  >>
>
> Agreed.  I don't get it ...    ?:-/
>
> 'Sorry if this has already been addressed ... but is there any 
> significance to the differences in the PARI code given for A124691 and 
> A124692, i.e.,
>
>    digitpos2ton(n) = \ 2^n expansion first occurrence of n { local(x, y, 
> r, dot); for(x=0, n, r = 2^x; ...
> vs.
>    digitposmton(n, m) = \ m^n expansion first occurrence of n { local(x, 
> y, r, dot); for(x=0, n, r = m^x; ...
>
> (Does the PARI code for either one yield a sequence anything like the 
> enigmatic "-1,-1,-1,-1,-1,-1,1,-1,-1,-1,1,4, ..."?)
>
> (For what it's worth, 51^11 = 6071163615208263051, which has the substring 
> "11" starting at the 4th digit, and 51 is the smallest positive integer 
> for which this is the case.)
>
> -- Jon
>
> ----- Original Message ----- 
> From: "N. J. A. Sloane" <njas at research.att.com>
> To: <seqfan at ext.jussieu.fr>
> Cc: <hillcino368 at hotmail.com>; <njas at research.att.com>
> Sent: Saturday, June 23, 2007 3:03 PM
> Subject: help needed with a duplicate
>
>
>> Dear Seqfans,  Andrew P. pointed out that A124691 and A124692
>> were the same.  So I tried to edit A124692.
>> The old entry was :
>>
>> %I A124692
>> %S A124692 
>> 1,1,1,1,1,1,1,1,1,1,1,4,4,4,5,5,5,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,10,11,3,
>> %T A124692 
>> 10,2,12,12,13,13,13,13,10,14,14,15,15,5,16,3,16,16,17,17,17,18,18,18,11,19,19,19,20,
>> %U A124692 
>> 20,20,13,21,21,22,22,2,5,23,23,23,24,24,24,25,25,20,25,26,4,26,27,27,3,28,28,28,24,29
>> %V 
>> A124692 -1,-1,-1,-1,-1,-1,1,-1,-1,-1,1,4,4,4,5,5,5,6,6,6,7,7,7,7,8,8,8,9,9,9,10,10,10,10,11,3,
>> %W A124692 
>> 10,2,12,12,13,13,13,13,10,14,14,15,15,5,16,3,16,16,17,17,17,18,18,18,11,19,19,19,20,
>> %X A124692 
>> 20,20,13,21,21,22,22,2,5,23,23,23,24,24,24,25,25,20,25,26,4,26,27,27,3,28,28,28,24,29
>> %N A124692 Position of the first n in the expansion of 2^n, or -1 if the 
>> digits of n never appear.
>> %o A124692 (PARI) digitposmton(n,m) = \ m^n expansion first occurrence of 
>> n { local(x,y,r,dot); for(x=0,n, r = m^x; if(r==floor(r), 
>> y=find(Str(floor(r)),x), y=find(Str(r),x); dot=find(Str(r),"."); if(dot < 
>> y, y--); ); if(y, print1(y","),print1(-1",") ) )
>> find(str,match) = \Return the position of the first occurrence of string 
>> \match in string str { local(lnm,lns,x,c,i); str=Str(str); \This allows 
>> leaving quotes off input match=Str(match); c=0; i=0; lns=length(str); 
>> lnm=length(match); if(lnm>1,i=1); x=1;
>> if(x>lns,return(0),return(x)) } mid(str,s,n) = \ Get a substring of 
>> length n from string str starting at position s in str. { 
>> local(v,ln,x,tmp); v =""; tmp = Vec(str); ln=length(tmp); for(x=s,s+n-1, 
>> v=concat(v,tmp[x]); ); return(v) }
>> %K A124692 base,easy,sign
>> %O A124692 0,12
>> %A A124692 Cino Hilliard (hillcino368(AT)hotmail.com), Dec 25 2006
>>
>> I tried to edit the description to the following:
>>
>> %N A124692 Starting position of the first n in the decimal expansion of 
>> 2^n, reading the decimal string "2^n" from left to right; or -1 if the 
>> digits of n never appear.
>>
>> and added some examples:
>>
>> %e A124692 2^6 = 64, where "6" appears in position 1, so a(6) = 1.
>> %e A124692 2^10 = 1024, so a(10) = 1.
>> %e A124692 But why is a(11) = 4?
>>
>> it seems to me a(11) = -1 since 2^11 = 2048
>>
>> What am I missing?
>>
>> Neil
>>
>>
>
>
>
>
> -- 
> No virus found in this incoming message.
> Checked by AVG Free Edition. Version: 7.5.472 / Virus Database: 
> 269.9.6/863 - Release Date: 6/23/2007 11:08 AM
> 

David Wilson wrote:

> %N A124692 n is a substring of of 2^A000000(n) starting at the a(n) 
> th digit from the left.
> %S A124692  
> 1,1,3,10,2,10,5,3,11,13,2,5,20,4,3,24,19,5,17,19,35,25,53,67,60,61,33, 
> 74,18,109,

This approach, at least as defined above, has it's own drawback: The  
as well as 5.