%C A136296 "Special augmented numbers" from Zak Seidov

[franktaw at netscape.net]

This is not so difficult.  We're looking for k digit numbers that 
divide
10^{k+1}+1.  (Anybody not see why this is equivalent?)

Some PARI:

A136296k(k) = {local(l, d, lb, ub);
 d=divisors(10^(k+1)+1);l=[];
 lb=10^(k-1);
 ub=10*lb;
 for(i=1,#d,if(d[i]>=lb&&d[i]<ub,l=concat(l,[d[i]])));
 l}

l=[];for(i=1,20,l=concat(l,A136296k(i)));l

And we get:

1, 11, 13, 77, 91, 137, 9091, 909091, 5882353, 10989011,
12987013, 52631579, 76923077, 90909091, 4347826087,
9090909091, 13698630137, 909090909091, 3448275862069,
10989010989011, 12987012987013, 76923076923077,
90909090909091, 9090909090909091, 909090909090909091,
1369863013698630137, 10989010989010989011,
12987012987012987013, 20408163265306122449,
76923076923076923077, 90909090909090909091

Franklin T. Adams-Watters

-----Original Message-----
From: Maximilian Hasler <maximilian.hasler at gmail.com>

1p1 divisible by p
<=> 10^[log10(p)+2] + 10p + 1 divisible by p
<=> 10^[log10(p)+2] + 1 divisible by p
<=> 10^[log10(p)+2] = -1 (mod p)

Three more terms ; a(7)>10^7.

%I A136296
%S A136296 11, 13, 137, 9091, 909091, 5882353
%F A136296 1p1 divisible by p <=> 10^[log[10](p)+2] = -1 (mod p)
%o A136296 (PARI) forprime(p=1,10^7,Mod(10,p)^(log(p)\log(10)+2)+1 |
print1(p", "))
%A A136296 M. F. Hasler (Maximilian.Hasler at gmail.com), Apr 22 2008


On Tue, Apr 22, 2008 at 1:24 AM, zak seidov <zakseidov at yahoo.com> wrote:
> %C A136296 "Special augmented numbers"  p such that
>  the decimal number 1p1 is divisible by p:
>  
1,11,13,77,91,137,9091,909091,90909091,9090909091,909090909091,9090909090
9091,9090909090909091,909090909090909091.
>  Notice the infinite pattern
>  p=(90..90..90)91 with 1p1/p=21, e.g.,
>  1911/91=190911/9091=19090911/909091=21.
>
>  Prime numbers are
>  11, 13, 137, 9091, 909091, 909090909090909091








I apologize for my misinterpretation, which was pointed out to me by David 
Wilson below. For the avoidance of doubt:


I don't think one should put in "conjectured" zeros.

Drew

drew at math.mit.edu pisze:
> You are right. I interpreted Artur's suggestion as a(n)=0 if no such
> prime exists, but on re-reading I realize this is not quite what he
> said. I totally agree that a(n)=0 if a(n) is unknown is a bad idea.
> 
> Is there a way to put in a placeholder (e.g. a ? or *) for unknown
> terms of a sequence so that one can at least enter known values for
> later terms without having to list them all in the comments?
> 
> Drew
> 
> On Apr 22 2008, David W. Wilson wrote:
> 
>> Bad idea. Neil is OK with "a(n) = 0 if a(n) does not exist" but is
>> strongly
>> averse to "a(n) = 0 if a(n) is unknown".
>> 
>>> -----Original Message-----
>>> From: drew at math.mit.edu [mailto:drew at math.mit.edu]
>>> Sent: Tuesday, April 22, 2008 6:48 AM
>>> To: grafix at csl.pl
>>> Cc: seqfan; njas at research.att.com
>>> Subject: Re: A139074 or 0 if no such prime exists.
>>> 
>>> While I am not as pessimistic as Artur on the Fermat primes question
>>> (or
>>> maybe I just hope for a long life :)), I agree that this is a better
>>> definition.
>>> 
>>> Drew
>>> 
>>> On Apr 22 2008, Artur wrote:
>>> 
>>> > Dear Neil,
>>> > Proof that such prime in A139074 doesn't exist will be as difficult as
>>> > proof that set of Fermat Primes is complete, and surely we don't see
>>> > these proof for our life. Mayby will be better to say  0 if we don't
>>> > know yet such prime.
>>> > Best wishes
>>> > Artur
>>> > On Apr 22 2008, franktaw at netscape.net wrote:

>I disagree.  The definition of the sequence should not depend upon the
>state of our knowledge.
>
>Instead, a comment should be added to note that the 0's in the sequence
>are conjectured.
>
>Franklin T. Adams-Watters
>
>-----Original Message-----
>From: drew at math.mit.edu
>
>While I am not as pessimistic as Artur on the Fermat primes question 
>(or
>maybe I just hope for a long life :)), I agree that this is a better
>definition.
>
>Drew
>
>On Apr 22 2008, Artur wrote:
>
>>Dear Neil,
>
>>Proof that such prime in A139074 doesn't exist will be as difficult as
>>proof that set of Fermat Primes is complete, and surely we don't see
>>these proof for our life. Mayby will be better to say  0 if we don't
>>know yet such prime.
>
>>Best wishes
>>Artur
>