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...
---------------------------------
Here is the
construction algorithm:
1) Lots of dots
(“holes”):
S = . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
2) Start with
integers 1 and 2:
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.
- -
-
Breaking
News (oct. 15th, 2007): Maximilian Hasler asks me if 2007 = a(14868). I’m afraid I don’t
know :-/
Back to a certain page, here.