A fractal sequence with prime sums

S = 1 2 1 2 3 4 5 1 2 6 3 4 7 8 5 9 1 2 10 6 3 4 11 7 8 5 12 13 9 1 2 14 10 6 3 4 15 16 17 11 7 8 5 18 19 20 12 13 9 1 2 21 22 23 14 24 10 6 3 4 25 26 15 27 16 28 17 11 7 8 5 29 18 30 19 20 12 13 9 1 2 31 21 22 32 23 14 33 34 35 24 36 10 6 3 4 37 38 25 26 15 39...

This sequence is fractal if you “upper trim” it (mark in yellow the first occurrence of “1”, then the first “2”, the first “3”, the first “4”, etc. -- i.e. the natural numbers); the non yellowed terms are the sequence itself:

S = 1 2 1 2 3 4 5 1 2 6 3 4 7 8 5 9 1 2 10 6 3 4 11 7 8 5 12 13 9 1 2 14 10 6 3 4 15 16 17 11 7 8 5 18 19 20 12 13 9 1 2 21 22 23 14 24 10 6 3 4 25 26 15 27 16 28 17 11 7 8 5 29 18 30 19 20 12 13 9 1 2 31 21 22 32 23 14 33 34 35 24 36 10 6 3 4 37 38 25 26 15 39...

This sequence is also fractal if you look at it from another point of view.

Rule: if, in an (a,b,c) triplet of consecutive terms a+b is prime, then mark c in yellow:

S = 1 2 1 2 3 4 5 1 2 6 3 4 7 8 5 9 1 2 10 6 3 4 11 7 8 5 12 13 9 1 2 14 10 6 3 4 15 16 17 11 7 8 5 18 19 20 12 13 9 1 2 21 22 23 14 24 10 6 3 4 25 26 15 27 16 28 17 11 7 8 5 29 18 30 19 20 12 13 9 1 2 31 21 22 32 23 14 33 34 35 24 36 10 6 3 4 37 38 25 26 15 39...

Voilà, the non yellowed terms rebuild also the sequence.

Different rules, same result. I just wanted to sow as chaotically as possible the “upper trimmed” integers in the sequence...

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Here is the construction algorithm:

1) Lots of dots (“holes”):

S = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

S = 1 2  . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3) Last two terms sum to a prime, so next term is in “yellow”:

S = 1 2 . . . . . . . . . . . . . . .  . . . . . . . . . . . . .

4) Sow in the first empty yellow hole the smallest natural number not yet in yellow:

S = 1 2 1  . . . . . . . . . . . . . . . . . . . . . . . . . . .

5) Last two terms sum to a prime, next term is in yellow:

S = 1 2 1 . . . . . . . . . . . . . .  . . . . . . . . . . . . .

6) Sow in the first empty yellow hole the smallest natural number not yet in yellow:

S = 1 2 1 2  . . . . . . . . . . . . . . . . . . . . . . . . . .

7) Last two terms sum to a prime, next term is in yellow:

S = 1 2 1 2 . . . . . . . . . . . . . . . . . . . . . . . . .  .

8) Sow in the first empty yellow hole the smallest natural number not yet in yellow:

S = 1 2 1 2 3  . . . . . . . . . . . . . . . . . . . . . . . . .

9) Last two terms sum to a prime, next term is in yellow:

S = 1 2 1 2 3 . . . . . . . . . . . . . . . . . . . .  . . . . .

10) Sow in the first empty yellow hole the smallest natural number not yet in yellow:

S = 1 2 1 2 3 4  . . . . . . . . . . . . . . . . . . . . . . . .

11) Last two terms sum to a prime, next term is in yellow:

S = 1 2 1 2 3 4 . . . . . . . . . . . . . . . . . . .  . . . . .

12) Sow in the first empty yellow hole the smallest natural number not yet in yellow:

S = 1 2 1 2 3 4 5  . . . . . . . . . . . . . . . . . . . . . . .

13) Last two terms sum NOT to a prime, next term is the 3rd term:

S = 1 2 1 2 3 4 5 1 . . . . . . . . . . . . . . . . . . . .  . .

13) Last two terms sum NOT to a prime, next term is the 4th term:

S = 1 2 1 2 3 4 5 1 2  . . . . . . . . . . . . . . . . . . . . .

14) Last two terms sum to a prime, next term is in yellow:

S = 1 2 1 2 3 4 5 1 2 . . . . . . . . . . . . . . . . . . .  . .

15) Sow in the first empty yellow hole the smallest natural number not yet in yellow:

S = 1 2 1 2 3 4 5 1 2 6  . . . . . . . . . . . . . . . . . . . .

16) Last two terms sum NOT to a prime, next term is the 5th term:

S = 1 2 1 2 3 4 5 1 2 6 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17) Last two terms sum NOT to a prime, next term is the 6th term:

S = 1 2 1 2 3 4 5 1 2 6 3 4 . . . . . . . . . . . . . . .  . . .

18) Last two terms sum to a prime, next term is in yellow:

S = 1 2 1 2 3 4 5 1 2 6 3 4  . . . . . . . . . . . . . . . . . .

19) Sow in the first empty yellow hole the smallest natural number not yet in yellow:

S = 1 2 1 2 3 4 5 1 2 6 3 4 7 . . . . . . . . . . . . . .  . . .

20) Last two terms sum to a prime, next term is in yellow:

S = 1 2 1 2 3 4 5 1 2 6 3 4 7  . . . . . . . . . . . . . . . . .

21) Sow in the first empty yellow hole the smallest natural number not yet in yellow:

S = 1 2 1 2 3 4 5 1 2 6 3 4 7 8 . . . . . . . . . . . . .  . . .

22) Last two terms sum NOT to a prime, next term is the 7th term:

S = 1 2 1 2 3 4 5 1 2 6 3 4 7 8 5  . . . . . . . . . . . . . . .

23) Last two terms sum to a prime, next term is in yellow:

S = 1 2 1 2 3 4 5 1 2 6 3 4 7 8 5 . . . . . . . . . . . .  . . .

24) Sow in the first empty yellow hole the smallest natural number not yet in yellow:

S = 1 2 1 2 3 4 5 1 2 6 3 4 7 8 5 9  . . . . . . . . . . . . . .

25) Last two terms sum NOT to a prime, next term is the 8th term:

S = 1 2 1 2 3 4 5 1 2 6 3 4 7 8 5 9 1 . . . . . . . .  . . . . .

etc.

__________

At what index does 2007 appear?

;-)

Best,

É.

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Breaking News (oct. 15th, 2007): Maximilian Hasler asks me if 2007 = a(14868). I’m afraid I don’t know :-/

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