An array alternating non-primes

and primes

 

 

 

[Sent: Monday, April 8, 2013 0:00 AM]

 

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 my 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 & content

 

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 sequence is of course a permutation of the natural numbers.

 

Best,

….

 

__________

 

[Maximilian Hasler]:

 

Dear Eric & SeqFans,

 

First (yet somehow least important), let me just correct a smallerror, "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 thereíd 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,

...

 

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

 

__________

 

[Hans Havermann]:

 

Maximilian Hasler:

 

> 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,

> ...

 

If you just want to see more rows, Iíve put another 150 here:

 

http://chesswanks.com/num/NotTriangular/

 

nt_1.gifnt_2.gif

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

39

40

42

44

45

46

48

49

50

149

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157

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191

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51

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197

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257

263

269

271

66

68

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85

277

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293

307

311

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331

337

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359

367

373

86

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105

106

379

383

389

397

401

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421

431

433

439

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449

457

461

463

467

479

487

108

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111

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119

120

121

122

123

124

125

126

128

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132

133

134

491

499

503

509

521

523

541

547

557

563

569

571

577

587

593

599

601

607

613

617

619

631

641

643

647

653

659

661

673

135

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150

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677

683

691

701

709

719

727

733

739

743

751

757

761

769

773

787

797

809

811

821

823

827

829

839

853

857

859

863

877

881

883

887

907

911

919

929

937

174

175

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182

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221

941

947

953

967

971

977

983

991

997

1009

1013

1019

1021

1031

1033

1039

1049

1051

1061

1063

1069

1087

1091

1093

1097

1103

1109

1117

1123

1129

1151

1153

1163

1171

1181

1187

1193

1201

1213

1217

1223

1229

1231

222

224

225

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230

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245

1237

1249

1259

1277

1279

1283

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1301

1303

1307

1319

1321

1327

1361

1367

1373

1381

1399

246

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1409

1423

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1447

1451

1453

1459

1471

1481

1483

1487

1489

1493

1499

1511

1523

1531

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1549

272

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299

 

__________

 

Many thanks, Maximilian and Hans!

Best,

….