# [seqfan] Re: A sequence based on ?ukasiewicz Logics and Prime Numbers

Artur grafix at csl.pl
Sun Feb 6 11:40:20 CET 2011

```Generally definition isn't as easy.

Chapter VI A matrix logic for prime numbers and the law of generation of
prime numbers

Every prime belonging to the one and only one class i

P_i where i is class number

Algorhithm is complicate includes iterractive logical procedures with
use Schaeffer stroke. Author used to this purpose especiall computer
programme of V.I. Shalack

P0 is set { 3, 5, 7, 11, 13}
P1 is P0 ∪ {17, 19},
P2 = P1 ∪ {23, 29, 31, 41, 43, 53, 59, 61}
These three above was discovered(counted) Karpenko 1982
P3 = P2 ∪ {37, 47, 109}
P4=P3 ∪ {...}
etc.

Theorem 7. Every odd prime number is contained in some class Pi

We can construct also new sequences for ONEIS:

Smallest prime p contained to the new class i
3,17,23,37, ... new for ONEIS

HYPOTHESIS3. Every class Pi is finite

consequence this hypethesis is next

HYPOTHESIS (Jasinski) Number of class Pi is infinite

If this hypothesis is true existed infinite sequence

Biggest prime p contained to the new class i
13,19,61,109, ... new for ONEIS

Best wishes
Artur

W dniu 2011-02-06 03:55, Brendan McKay pisze:
> Some pages of this book can be read at Google Books (logging
> in might be necessary).  This sequence appears to be the one
> in Table 3 that starts on page 135 (many terms are given).
> The table caption is "Values of function i(p): classes of
> prime numbers".  It is explained on pages 99 onwards but
> I don't have time to decode it and I can't see all the
> pages of that section.
>
> Brendan.
>
>> Message: 15
>> Date: Sat, 5 Feb 2011 15:08:52 -0500
>> From: Charles Greathouse<charles.greathouse at case.edu>
>> Subject: [seqfan] A sequence based on ?ukasiewicz Logics and Prime
>> 	Numbers
>> To: Sequence Fanatics Discussion list<seqfan at list.seqfan.eu>
>> Message-ID:
>> 	<AANLkTikTDOALnpbKtQvcnrAwC=aixUsN+fvtY3PhJ8-e at mail.gmail.com>
>> Content-Type: text/plain; charset=UTF-8
>>
>> Does anyone have the book ?ukasiewicz Logics and Prime Numbers by A S
>> Karpenko?  There's a 'mystery sequence' A173883 which is based on that
>> book and it would be nice to have a definition!  My library system
>> doesn't have a copy (and Worldcat says the nearest copy is 300 miles
>> away).
>>
>> For those who speak Russian the original Logiki Lukasevica i prostye
>> cis is in that language.
>>
>> Charles Greathouse
>> Analyst/Programmer
>> Case Western Reserve University
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```