[seqfan] Re: zig-zag pseudoprimes

Richard Mathar mathar at strw.leidenuniv.nl
Wed Sep 1 12:54:33 CEST 2010

```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

```