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 A171100 Natural numbers arranged into triples a, b, c with no common factor. 2

%I

%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,

%T 43,20,31,51,22,39,61,24,41,65,26,33,59,28,45,73,30,47,77,32,49,81,34,

%U 53,87,36,55,91,38,63,101,40,57,97,42,67,109,44,69,113,46,71,117,48,79

%N Natural numbers arranged into triples a, b, c with no common factor.

%C Regard the sequence S as a succession of triple [a,b,c]:

%C 1,2,3,

%C 4,5,9,

%C 6,7,13,

%C 8,11,19,

%C 10,17,27,

%C 12,23,35,

%C 14,15,29,

%C 16,21,37,

%C 18,25,43,

%C ...

%C Rule 1) a+b=c

%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)

%C Rule 3) S is a permutation of the natural numbers.

%C To build S is easy:

%C - write down N

%C - start from the left and:

%C -> put a "+" on top of two as yet unmarked integers which will satisfy rules (1) and (2) (always start with the smallest unmarked integer)

%C -> put a "=" on top of the result taking the same rules into account

%C We have:

%C 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 ...

%C ... + + =

%C 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 ...

%C giving the first triple [1,2,3]

%C (used integers will be marked with a circle "o" from now on)

%C ....o.o.o.+.+.......=

%C 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 ...

%C giving the second triple [4,5,9]

%C ....o.o.o.o.o.+.+...o..........=

%C 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 ...

%C giving the third triple [6,7,13]

%C ....o.o.o.o.o.o.o.+.o....+.....o.................=

%C 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 ...

%C giving the fourth triple [8,11,19]

%C ....o.o.o.o.o.o.o.o.o.+..o.....o...........+.....o.......................=

%C 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 ...

%C giving the fifth triple [10,17,27]

%C ....o.o.o.o.o.o.o.o.o.o..o..+..o...........o.....o...........+............o......................=

%C 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 ...

%C giving the sixth triple [12,23,35]

%C etc.

%C Is the sequence infinite?

%C Contribution from _William Rex Marshall_, Nov 07 2010: (Start)

%C Sequence is infinite. A new triple is always possible as there are always infinitely many candidate "b" values coprime to the smallest unused integer "a", and previous triples can only rule out a finite number of them and their sums (which are necessarily pairwise coprime to "a" and "b").

%C It appears that the ratios a(n-2):a(n-1):a(n), when n is a multiple of 3, tend to 1:phi:phi^2 as n tends to infinity, where phi is the golden ratio (A001622). Is there a simple proof of this? (End)

%H W.R. Marshall, <a href="/A171100/b171100.txt">Table of n, a(n) for n=1..10000</a>

%H Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/TripletsPermut.htm">Naturals permuted in triple a, b, c "with no common factor"</a>

%H E. Angelini, <a href="/A171100/a171100.pdf">Naturals permuted in triple a, b, c "with no common factor"</a> [Cached copy, with permission]

%Y Cf. A171101 (values of c).

%K nonn,tabf

%O 1,2

%A _N. J. A. Sloane_, Sep 24 2010, based on a posting by Eric Angelini to the Sequence Fans Mailing List, Sep 16 2010

%E More terms from _R. J. Mathar_, Sep 25 2010

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Last modified May 15 10:54 EDT 2021. Contains 343909 sequences. (Running on oeis4.)