[seqfan] Re: help needed editing A175625

Sun Aug 1 17:30:32 CEST 2010

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
> 
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> 
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> 

 Shevelev Vladimir‎