# [seqfan] Naturals permutated in triplets a, b, c "with no common factor"

Eric Angelini Eric.Angelini at kntv.be
Thu Sep 16 15:39:21 CEST 2010

```Hello SeqFans,

(mail also available here:
http://www.cetteadressecomportecinquantesignes.com/TripletsPermut.htm)

S = 1,2,3,4,5,9,6,7,13,8,11,19,10,17,27,12,23,35,14,15,29,16,21,37,18,25,43,...

See S as a succession of triplets [a,b,c]:

S =
1,2,3,
4,5,9,
6,7,13,
8,11,19,
10,17,27,
12,23,35,
14,15,29,
16,21,37,
18,25,43,
...

Rule 1) a+b=c
Rule 2) "a" and "b" share no common factor (except 1)
"b" and "c" share no common factor (except 1)
"c" and "a" share no common factor (except 1)
Rule 3) S is a permutation of the Naturals

To build S is easy:
- write N
- start from the left and:
-> put a "+" on top of two yet unmarked integer which will obey
rules (1) and (2) (always start with the smallest unmarked integer)
-> put a "=" on top of the result taking the same rules into account

We have:

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

+ + =
N = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ...
giving the first triplet [1,2,3]
(used integers will be marked with a dot "." from now on)

. . . + +       =
N = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ...
giving the second triplet [4,5,9]

. . . . . + +   .          =
N = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ...
giving the third triplet [6,7,13]

. . . . . . . + .    +     .                 =
N = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ...
giving the third triplet [8,11,19]

. . . . . . . . . +  .     .           +     .                       =
N = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ...
giving the fourth triplet [10,17,27]

. . . . . . . . . .  .  +  .           .     .           +            .                      =
N = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ...
giving the sixth triplet [12,23,35]

etc.

I'm not 100% sure that S is infinite...
Could someone please compute a hundred or so terms (if of interest)
The "c" terms succession migth also be of interest.
Best,
É.

```

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