# [seqfan] Re: zig-zag pseudoprimes

Wed Sep 1 16:19:27 CEST 2010

```Thanks, Richard!

Nevertheless, I did not understand if you really obtained that, e.g., A000111(561)==1(mod 561)?

Regards,

----- Original Message -----
From: Richard Mathar <mathar at strw.leidenuniv.nl>
Date: Wednesday, September 1, 2010 14:29
Subject: [seqfan] Re: zig-zag pseudoprimes
To: seqfan at seqfan.eu

>
> Followup on http://list.seqfan.eu/pipermail/seqfan/2010-
> September/005897.html :
>
> > Recently I proved (not basing on Fermat little theorem)
> that  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).
> > I call possible composite numbers m with such property "zig-
> zag pseudoprimes".
>
> Numbers n such that A000111(n) == 1 mod (n) are
>
> 2,4,5,6,8,10,13,14,16,17,22,26,29,30,32,34,37,38,41,46,53,58,61,62,64,73,74,82,86,
> 89,94,97,101,106,109,113,118,122,128,134,137,142,146,149,157,158,166,173,178,181,
> 182,193,194,197,202,206,214,218,226,229,233,241,254,256,257,262,269,274,277,278,281,
> 293,298,302,313,314,317,326,334,337,346,349,353,358,362,373,382,386,389,394,397,398
>
> Intersecting with A016813 (4n+1), then removing the set of
> A002144 (primes 4n+1),
> the composites 561, 781, 1105... remain (which one can look up
> in the OEIS,
> perhaps A122782, A020142, A020174 etc).
>
> For the case of composites n == 3 ( mod 4) with A000111(n) == -1
> (mod n)
> we have at least the case n = 91 (no other <= 850)
>
> This is all subject to checking by an independent committee. I
> can submit them,
> but I shall not. (I do not submit sequences invented by other people.)
>
> Richard Mathar
>
>
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