# [seqfan] Re: zig-zag pseudoprimes

Fri Sep 3 09:31:22 CEST 2010

```I was successful to complete a proof of the following more general statement:

1) if n==1(mod 4) is prime, then  A000111(n)==1(mod n); if n==3(mod 4) is prime, then A000111(n)==-1(mod n);
2) if n is a positive power of 2, then  A000111(n)==1(mod n);
3) if n is double odd prime, then  A000111(n)==1(mod n).

The latter result allows to introduce "zig-zag pseudo-double-primes". They are double odd composite
numbers n, for which we have A000111(n)==1(mod n).
The sequence of them begins from 30, 182.
I think that it is also rather interesting sequence for the consideration.

Regards,

----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Thursday, September 2, 2010 16:08
Subject: [seqfan] Re: zig-zag pseudoprimes
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> 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) 4*k+1-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
> > 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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> >
>