[seqfan] Re: A (not triangular) array inspired by A083197
Maximilian Hasler
maximilian.hasler at gmail.com
Mon Apr 8 16:50:46 CEST 2013
Dear Eric & SeqFans,
First (yet somehow least important), let me just correct a small
error, "51" is not prime and so the row starting with 47 should read
47,53,59,61,67,71,73.
Your post inspired me some more ideas:
First, I noticed that your construction can be iterated.
The first lines remain the same, but then, due to variing row lengths,
primes and non-primes get mixed in different ways.
But the changes appear later and later: In the next step, the sequence
would differ due to the row with length 16 instead of 17,
and the index of that row is the sum of all preceding numbers (way over 100).
Nonetheless, there is the "limiting" sequence to which the
construction converges (which coincides for the abovementioned reason
with the (corrected) terms of your example,
1,
2,3,
4,6,8,
5,7,11,13,
9,10,12,14,15,16,
17,19,23,29,31,37,41,43,
18,20,21,22,24,
47,53,59,61,67,71,73,
25,26,27,28,30,32,33,34,35,36,38,
79,83,89,97,101,103,107,109,113,127,131,137,139,
...
The same definition as A083197, but filling with nonnegative instead
of positive integers, yields another variation not in the OEIS:
0
2 3
1 4 6
5 7 11 13
8 9 10 12 14
(The odd rows are just "shifted" by 1 element wrt A083197, due to the
initial 0.)
To apply your idea here, we could say that there'd be an initial row 0
with 0 primes, so this empty row zero would be followed by
row 1 with 2 non-primes : 0, 1,
row 2 with 3 primes : 2,3, 5
row 3 with 1 non-prime : 4
row 4 with 4 primes : 7,11,13,17
row 5 with 6 composites : 6, 8, 9, 10, 12, 14
etc.
Iterating this construction once more gives:
row 0 with 0 primes,
row 1 with 1 non-primes : 0,
row 2 with 2 primes : 2,3,
row 3 with 3 non-primes : 1,4,6
row 4 with 5 primes : 5,7,11,13,17
row 5 with 4 composites : 8, 9, 10, 12
etc.
I think that even and odd sequences of this sequence (of sequences)
converge respectively to two distinct limits
0,
2,3,
1,4,6,
5,7,11,13,17,
8,9,10,12,
19,23,29,31,37,41,43,
14,15,16,18,20,21,22,24,25,26,27,
47,53,59,61,67,71,73,79,83,89,97,101,103,
28,30,32,33,34,35,36,38,39,40,42,44,45,46,48,49,50,
...
and
0,1,
2,3,5,
4,
7,11,13,17,
6,8,9,10,12,14,
19,23,29,31,37,
15,16,18,20,21,22,24,
41,43,47,53,59,61,67,71,73,79,83,
25,26,27,28,30,32,33,34,35,36,38,39,40,
...
Maximilian
PS: For my records, I get this by iterating
pnp(%,,1)
with
{pnp(a,nnp=-1,f=0,na=[],maxrow=10)=np=1;for(n=1,min(maxrow,#a),for(j=1,a[n],
na=concat(na,if(bittest(n,0)==f,np=nextprime(np+1),until(!isprime(nnp++),);nnp));print1(na[#na]","));print);na}
While for your sequences (starting with 1) I have to set the 2nd
parameter nnp to 0 and the 3rd (hack to exchange n even<=>odd) to 0
On Mon, Apr 8, 2013 at 9:14 AM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
> Hello SeqFans,
>
> http://oeis.org/A083197
> ... has given me the idea of a variation. Here is A083197:
>
>> Triangular array, read by rows, where
> if n is odd the n-th row consists of n least unused non-primes, while
> if n is even the n-th row consists of the n least unused primes.
>
> Triangle begins:
> 1
> 2 3
> 4 6 8
> 5 7 11 13
> 9 10 12 14 15
> 17 19 23 29 31 37
> etc.
> -----------------
>
> ... In this variation the size of each row is given by the successive
> integers of the sequence itself, not by n. Thus the array would begin:
>
> row size
> 1 1 (non-prime)
> 2 3 2 (primes)
> 4 6 8 3 (non-primes)
> 5 7 11 13 4 (primes)
> 9 10 12 14 15 16 6 (non-primes)
> 17 19 23 29 31 37 41 43 8 (primes)
> 18 20 21 22 24 5 (non-primes)
> 47 51 53 59 61 67 71 7 (primes)
> 25 26 27 28 30 32 33 34 35 36 38 11 (non-primes)
> ...
>
> As A083197, this new seq is of course a permutation of the natural numbers.
>
> Best,
> É.
>
>
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>
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