[seqfan] Re: A December quiz with a non-prime array

[Moshe Levin]

Happy Near Year!

Version of non-prime array.

 Triangle array biult and read by antidiagonals: 

each horizontal and vertical neighbor pair sums to odd composite number.

1,8,14,7,13,11,2,20,22,4,19,25,5,17,21,6,26,10,16,18,12,3,9,23,29,33,15,27,24,
30,40,28,36,42,48,38,31,39,35,37,41,45,43,47,49,32,46,52,50,44,54,72,34,68,56,
53,59,65,67,55,51,57,61,77,73,63,58

Triangle starts
  1, 8,  7,  2,19
14,13,20,25
11,22,5,
4,17
21

We start with 1. Then smallest integer m such that 1+m is odd composite is m=8.
Then smallest integer m (not used already) such that 1+m is odd composite is m=14.
Two antidiagonals are get. Now we proceed with 3rd antidiagonal.
Smallest integer m (not used already) such that 8+m is odd composite is m=7.
Then smallest integer m (not used already) such that both 8+m and 14+m are odd composite is m=13.
At last, smallest integer m (not used already) such that 14+m is odd composite is m=11. Third antidiagonal
is filled, etc.

First antidiagonals
{1},{8,14},{7,13,11},{2,20,22,4},{19,25,5,17,21}

And sequence is get by merging diagonals
1,8,14,7,13,11,2,20,22,4,19,25,5,17,21 

ML


28 декабря 2011, 23:23 от Maximilian Hasler <Maximilian.Hasler at martinique.univ-ag.fr>:
> Hello Eric,
> 
> I confirm your calculations and created the sequence http://oeis.org/A203071 ;
> the first row (now in oeis.org/A203072) goes on :
> [1, 8, 4, 14, 6, 10, 22, 27, 15, 35, 40, 88, 28, 26, 44, 60, 57, 58,
> 64, 120, 62, 72, 34, 56, 80, 63, 98, 46, 102, 178, 52, 100, 66, 156,
> 86, 82, 110, 76, 114, 94, 140, 96, 154, 130, 160, 112, 108, 138, 78,
> 150, 118, 146, 132, 126, 162, 176, 164, 188, 202, 200, 180, ...]
> 
> At that point there are 1891 numbers in the complete triangle, but the
> least composite not yet used is only 24;
> the 99 smallest numbers used so far are
> [1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28, 30,
> 32, 34, 35, 36, 38, 40, 42, 44, 46, 48, 49, 50, 51, 52, 54, 56, 57,
> 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84,
> 86, 88, 90, 91, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112,
> 114, 115, 116, 117, 118, 119, 120, 122, 124, 125, 126, 128, 129, 130,
> 132, 134, 136, 138, 140, 142, 143, 144, 146, 148, 150, 152, 154, 156,
> 158, 160, 161, 162]
> 
> Happy New Year,
> 
> Maximilian
> 
> PS: for my records:
> 
> extend_first_row( a=[], u=[] )={ u || for( i=1,#a, u=setunion(u,Set(a[i]));
>  forstep( j=i-1,1,-1, u=setunion(u, Set(a[j]+=a[j+1]))));
>  for( t=1,9e9, isprime(t) && next; setsearch(u,t) && next; my(tt=t);
>  forstep( j=#a,1,-1, setsearch(u, tt += a[j]) && next(2); isprime(tt)
> && next(2));
>  return(t)) }
> 
> list_by_antidiagonals(a)={my(u=[]);for(i=1,#a,u=concat(u,a[i]);forstep(j=i-1,1,-1,u=concat(u,a[j]+=a[j+1])));u}
> 
> On Wed, Dec 28, 2011 at 1:03 PM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
> > Hello Seqfans,
> > is it possible to put all non-primes in an array like
> > this one:
> > - every term is unique (and not prime)
> > - a term and his neighbor sum up on the line below.
> > Best,
> > É.
> > (hope no mistakes were left in the example below)
> >
> >           1    8    4     14     6     10    22     27   15
> >             9    12   18      20    16    32     49   42
> >               21   30     38     36    48     81    91
> >                 51    68     74     84    129   301
> >                   119 142 158 213   301
> >                      261    300    371    514
> >                         561     671    885
> >                            1232    1556
> >                                2788
> >
> > The array has been built by antidiagonals, choosing always
> > the smallest non-prime not yet in the array and not leading
> > to a contradiction.
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> 
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