[seqfan] Re: Pseudo-arithmetic progressions

Richard Mathar mathar at strw.leidenuniv.nl
Mon Jul 12 17:01:22 CEST 2010


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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