# [seqfan] Re: zig-zag pseudoprimes

Vladimir Shevelev shevelev at bgu.ac.il
Thu Sep 2 14:19:28 CEST 2010

I think, nevertheless, that your conclusion about the statistics of sequence {n: A000111(n)==1
(mod n)}, maybe, is not true for large n. Indeed, the 92 first terms of this sequence which you presented one can split into 4 subsequences: 1) odd primes; 2) positive powers of 2; 3) numbers of the form 2*p, where p is an odd prime; 4)  remaider subsequence {30,182,...}.
Till now I proved that  1) and 2) are indeed subsequences of the considered sequence and it is very plausible ( I did not yet prove this) that 3) is also subsequence. Since, nevertheless, union of these sequences has zero density, then your statement is true only in case when sequence {30,182,...} has a large density. It seems improbable.

Regards,

----- Original Message -----
From: Richard Mathar <mathar at strw.leidenuniv.nl>
Date: Wednesday, September 1, 2010 18:25
Subject: [seqfan] Re: zig-zag pseudoprimes
To: seqfan at seqfan.eu

>
> http://list.seqfan.eu/pipermail/seqfan/2010-
> September/005903.html :
> vs> Nevertheless, I did not understand if you really obtained
> that, e.g., A000111(561)==1(mod 561)?
>
>
> Yes. I naively trusted that the following Maple implementation
> works
> on my computer (one of the two branches of your problem in comments):
>
> A000111 := proc(n)
>         2^n*abs(
> euler(n,1/2)+euler(n,1)) ;
> end proc:
> for n from 1 do
>         if modp(A000111(n),n)
> = 1 and modp(n,4) = 1 and not isprime(n) then
>         # if
> modp(A000111(n),n) = n-1 and modp(n,4) = 3 and not isprime(n) then
>                 printf("%d,\n",n) ;
>         end if;
> end do:
>
>
> It is not unusual that A000111(n) == 1 (mod n), if we consider the
> statistics in the following sequence of A000111(n) mod n, n>=1:
>
> 0,1,2,1,1,1,6,1,7,1,10,5,1,1,2,1,1,7,18,5,19,1,22,17,16,1,2,5,1,1,30,1,31,1,
> 12,29,1,1,2,25,1,19,42,5,16,1,46,17,22,21
>
> This is the reason why the first sequence I showed in
> http://list.seqfan.eu/pipermail/seqfan/2010-September/005898.html
> is rather dense and filled four lines until reaching 400. 561
> comes later
> in the same sequence.
>
> RJM
>
>
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