# [seqfan] Re: Pseudo-arithmetic progressions

Mon Jul 12 19:15:56 CEST 2010

Very thanks, Richard,

You are right, I (calculating handy), indeed, omitted A179382(16)=1 and A179382(20)=4. You quite correctly understood A179383. Your so far calculations show that, most likely, \omega(2n-1) for the records is, indeed, 1 for n>1.

Regards,

----- Original Message -----
From: Richard Mathar <mathar at strw.leidenuniv.nl>
Date: Monday, July 12, 2010 18:26
Subject: [seqfan] Re: Pseudo-arithmetic progressions
To: seqfan at seqfan.eu

>
> Followup on http://list.seqfan.eu/pipermail/seqfan/2010-
> July/005279.html :
>
> v> Dear Seq Fans,
> v>
> v> My new submissions are:
> v>
> v> %I A179382
> v> %S A179382 1,1,2,1,3,5,6,1,4,9,2,4,10,9,14,5,5,18,10
> v> %N A179382 a(n) is the smallest period of pseudo-arithmetic
> progression with initial term 1 and difference 2n-1
>
> I get a different sequence with an additional 1 = a(14) and a 4
> after 18:
> 1, 1, 2, 1, 3, 5, 6, 1, 4, 9, 2, 4, 10, 9, 14, 1, 5, 5, 18, 4,
> 10, 7, 5,
> 9, 10, 2, 26, 8, 9, 29, 30, 1, 6, 33, 11, 14, 3, 9, 15, 17, 27,
> 41, 2, 11,
> 4, 4, 3, 14, 24, 15, 50, 23, 4, 53, 18, 14, 14, 19, 3, 9, 55, 6,
> 50, 1, 7,
> 65, 8, 17, 34, 69, 23, 25, 14, 20, 74, 5, 10, 8, 26, 21
>
> The records are
> 1, 2, 3, 5, 6, 9, 10, 14, 18, 26, 29, 30, 33, 41, 50, 53, 55,
> 65, 69, 74
> with record indices (positions) at
> n = 1, 3, 5, 6, 7, 10, 13, 15, 19, 27, 30, 31, 34, 42, 51, 54,
> 61, 66, 70, 75
>
> 2n-1 of the record positions n is
>
> 1,5,9,11,13,19,25,29,37,53,59,61,67,83,101,107,121,131,139,149,163,173,179,
> 181,197,211,227,269,293,317,347,349,373,379,389,419,421,443,461,467,491,509,
> 523,541,547,557,563,587,613,619,653,659,661,677,701,709,757
>
> which is A179383 if I understand this correctly.
>
> The small-omega (number of different primes) of these is 0
> followed by all-1
> if one includes terms up to A179382(680).
>
>
>
> v> %I A179383
> v> %S A179383 1,5,9,11,13,19,25,29,37
> v> %N A179383 Differences of pseudo-arithmetic progressions with
> initial term 1 (see A179382) for which the sequence of smallests
> periods is the sequence of records of A179382
> v> %C A179383 Question. Do exist terms of the sequence having
> more than 1 prime divisors?
> v> %Y A179383 A139099 A167791 A002326 A179382
> v> %K A179383 nonn
> v> %O A179383 1,2
>
> In Maple this is:
>
> A000265 := proc(n)
>         local d;
>
> numtheory[divisors](n) minus {seq(2*i,i=1..n/2)} ;
>         max(op(%)) ;
> end proc:
> pseuAprog := proc(a,b)
>         A000265(a+b) ;
> end proc:
> A179382 := proc(n)
>         local p,k;
>         p := [1] ;
>         for k from 2 do
>                 a := pseuAprog( p[-1],2*n-1) ;
>                 if not a in p then
>                         p := [op(p),a] ;
>                 else
>                         return nops(p) ;
>                 end if;
>         end do:
> end proc:
> seq(A179382(n),n=1..80) ;
>
>
>
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