[seqfan] Re: help needed editing A175625

Sun Aug 1 18:39:45 CEST 2010

Moreover, it is sufficient (with the same arguments) to find x such that A001567(x)=2*p+1, where p>3 is prime, and put m=(A001567(x)-7)/2.  It is easy to see that A001567(x) should be divisor of (2^p+1)*(2^p-1).

Vladimir

----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Sunday, August 1, 2010 18:58
Subject: [seqfan] Re: help needed editing A175625
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> Note that, if there exist x,y such that 
> A001567(x)=2*A001567(y)+1 and A001567(y) is not multiple of 3, 
> then in the sequence A175625 there exists a composite number. 
> Indeed,  number 
>  m=(A001567(x)-7)/2=(2*A001567(y)-6)/2
> satisfies all conditions: 1) it is not multiple of 3; 2) 
> 4^(m+3)=2^(A001567(x)-1)==1 (mod A001567(x)=1 (mod(2*m+7)); 3) 
> 4^(m+2)=2^(2*(A001567(y)-1))==1(mod A001567(y))=1 (mod (m+3)).
>  
> Since there exist many pairs of primes of the form (p, 2*p+1), 
> then it seems that nothing prevents to exist pairs of the form 
> (A001567(y), 2*A001567(y)+1) and, therefore most likely, one can 
> find composite numbers in A175625.
>  
> Vladimir
> 
> 
> ----- Original Message -----
> From: Vladimir Shevelev <shevelev at bgu.ac.il>
> Date: Sunday, August 1, 2010 12:57
> Subject: [seqfan] Re: help needed editing A175625
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> 
> > A definition is: Let m(n) be the n-th number such that 1) m(n)-
> 1 
> > is not multiple of 3, 2) 4^(m(n)+3)==1(mod(2*m(n)+7)), 
> > 3)4^(m(n)+2)==1 (mod (m(n)+3)). Then a(n)=2*m(n)+7.
> > 
> > E.g., m(1)=2 since 4^5==1(mod 11)  and 4^4==1(mod 5). 
> > Therefore, a(1)=2*m(1)+7=11.
> > 
> > Vladimir
> > 
> > ----- Original Message -----
> > From: "N. J. A. Sloane" <njas at research.att.com>
> > Date: Saturday, July 31, 2010 21:28
> > Subject: [seqfan]  help needed editing A175625
> > To: seqfan at seqfan.eu
> > Cc: njas at research.att.com
> > 
> > > Dear Seq Fans,
> > > Could someone please study this sequence and provide a 
> better 
> > > definition?Max Alekseyev, if you are around, it might help if
> > > you could ask the author in Russian exactly what the 
> > definition is.
> > > (I assume "compound" means "composite")
> > > 
> > > Thanks, Neil
> > > 
> > > The original sequence as submitted:
> > > 
> > > Date: Wed, 28 Jul 2010 09:51:59 -0400
> > > Subject: NEW SEQUENCE FROM submitN Alzhekeyev Ascar M A175625
> > > Reply-to: allasc at mail.ru
> > > 
> > > %I A175625
> > > %S A175625 11,23,31,47,59,83,107,167,179,227,263,347,359
> > > %N A175625 N(i)=2*i+7; ((2*i+7) mod 3)> 0; (N(i) does not 
> > share 
> > > on 3); 4 ^ (i+3) ��� 1 (mod (2*i+7)); 4 ^ (i+2) ��� 1 (mod 
> > > (i+3)) N(i) - primary number 
> > > %C A175625 N(i) - primary number 
> > > %C A175625 Then number N - idle time! It is checked up to N 
> > > <300 million. 
> > > %F A175625 N(i)=2*i+7; ((2*i+7) mod 3)> 0; (N(i) does not 
> > share 
> > > on 3); 4 ^ (i+3) ��� 1 (mod (2*i+7)); 4 ^ (i+2) ��� 1 (mod 
> > > (i+3)) 
> > > %F A175625 N(i) - primary number 
> > > %K A175625 nonn
> > > %O A175625 1,1
> > > %A A175625 Alzhekeyev Ascar M (allasc(AT)mail.ru), Jul 28 2010
> > > 
> > > My edited version (which probably did not make things better):
> > > 
> > > %I A175625
> > > %S A175625 11,23,31,47,59,83,107,167,179,227,263,347,359
> > > %N A175625 a(i)=2*i+7; ((2*i+7) mod 3)> 0; (a(i) does not 
> > share 
> > > on 3); 4^(i+3) == 1 (mod (2*i+7)); 4^(i+2) == 1 (mod (i+3)).
> > > %C A175625 Checked up to a <300 million.
> > > %K A175625 nonn,uned,new
> > > %O A175625 1,1
> > > %A A175625 Alzhekeyev Ascar M (allasc(AT)mail.ru), Jul 28 2010
> > > %E A175625 Partially edited by N. J. A. Sloane, Jul 29 2010
> > > 
> > > An update submitted via SubmitC.html:
> > > 
> > > Date: Fri, 30 Jul 2010 07:00:11 -0400
> > > Subject: COMMENT FROM submitC Alzhekeyev Ascar M A175625
> > > 
> > > START SUBMITC MESSAGE
> > > Sequence: A175625
> > > Sender: Alzhekeyev Ascar M
> > > Address: allasc at mail.ru
> > > Date of submission: Jul 30 2010
> > > 
> > > Description of changes, start:
> > > Description of changes, end
> > > 
> > > cdiff, start:
> > > *** /home/oeis/wwwfiles/sequences/tmp/tempC1.14292 Fri Jul 
> 30 
> > > 07:00:11 2010
> > > --- /home/oeis/wwwfiles/sequences/tmp/tempC2.14292 Fri Jul 
> 30 
> > > 07:00:11 2010
> > > ***************
> > > *** 2,7 ****
> > > --- 2,13 ----
> > >   %S A175625 
> > 11,23,31,47,59,83,107,167,179,227,263,347,359 
> > >   %N A175625 a(i)=2*i+7; ((2*i+7) mod 3)> 0; (a(i) 
> > does not 
> > > share on 3); 4^(i+3) == 1 (mod (2*i+7)); 4^(i+2) == 1 (mod 
> > > (i+3)). 
> > >   %C A175625 Checked up to a <300 million. 
> > > + %C A175625 30 Jul 2010 
> > > + %C A175625 The first compound number a(i)=536870911 
> > > + %C A175625 It is found by the user "venco" (from 
> > > http://dxdy.ru) 
> > > + %C A175625 It is interesting to notice that 
> > i+3=268435455=2^28-
> > > 1 
> > > + %C A175625 Please inform on a finding of following 
> compound 
> > > numbers in sequence. 
> > > + %C A175625 Properties of number i+3 at exceptions 
> (compound 
> > > numbers) are very interesting 
> > >   %K A175625 nonn,uned,new 
> > >   %O A175625 1,1 
> > >   %A A175625 Alzhekeyev Ascar M 
> (allasc(AT)mail.ru), 
> > Jul 28 
> > > 2010 
> > > cdiff, end
> > > 
> > > Also a huge b-file arrived (with a different offset).
> > > 
> > > The present version is as follows:
> > > 
> > > %I A175625
> > > %S A175625 11,23,31,47,59,83,107,167,179,227,263,347,359
> > > %N A175625 a(i)=2*i+7; ((2*i+7) mod 3)> 0; (a(i) does not 
> > share 
> > > on 3); 4^(i+3) == 1 (mod (2*i+7)); 4^(i+2) == 1 (mod (i+3)).
> > > %C A175625 Checked up to a <300 million.
> > > %C A175625 The first compound number a(i)=536870911
> > > %C A175625 It is found by the user "venco" (from http://dxdy.ru)
> > > %C A175625 It is interesting to notice that 
> i+3=268435455=2^28-1
> > > %C A175625 Please inform on a finding of following compound 
> > > numbers in sequence.
> > > %C A175625 Properties of number i+3 at exceptions (compound 
> > > numbers) are very interesting
> > > %H A175625 Alzhekeyev Ascar M, Table of n, a(n) for n = 
> 2..11941610> > %K A175625 nonn,uned,new
> > > %O A175625 1,1
> > > %A A175625 Alzhekeyev Ascar M (allasc(AT)mail.ru), Jul 28 
> > 2010, 
> > > Jul 30 2010
> > > %E A175625 Partially edited by N. J. A. Sloane, Jul 29 2010
> > > 
> > > 
> > > 
> > 
> >  Shevelev Vladimir‎
> > 
> > _______________________________________________
> > 
> > Seqfan Mailing list - http://list.seqfan.eu/
> > 
> 
>  Shevelev Vladimir‎
> 
> _______________________________________________
> 
> Seqfan Mailing list - http://list.seqfan.eu/
> 

 Shevelev Vladimir‎