

A127202


a(1)=1, a(2)=2; a(n) = the smallest positive integer not occurring earlier in the sequence such that gcd(a(n), a(n1)) does not equal gcd(a(n1), a(n2)).


10



1, 2, 4, 3, 6, 5, 10, 7, 14, 8, 9, 12, 11, 22, 13, 26, 15, 18, 16, 17, 34, 19, 38, 20, 21, 24, 23, 46, 25, 30, 27, 28, 32, 29, 58, 31, 62, 33, 36, 35, 40, 37, 74, 39, 42, 41, 82, 43, 86, 44, 45, 48, 47, 94, 49, 56, 50, 51, 54, 52, 53, 106, 55, 60, 57, 59, 118, 61, 122, 63, 66
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OFFSET

1,2


COMMENTS

This sequence appears to be a permutation of the positive integers.  Leroy Quet, Jan 08 2007
From N. J. A. Sloane, Jan 26 2017: (Start)
Theorem: This is a permutation of the positive integers.
Proof: (Outline. For details see the link.)
1. Sequence is infinite.
2. For all m, either m is in the sequence or there exists an n_0 such that for n >= n_0, a(n) > m.
3. For all primes p, there is a term divisible by p.
4. For all primes p, there are infinitely many multiples of p in the sequence.
5. Every prime appears in the sequence.
6. For any number m, there are infinitely many multiples of m in the sequence.
7. Every number m appears in the sequence.
(End)
Comment from N. J. A. Sloane, Feb 28 2017 (Start):
There are several short cycles and at least one apparently infinite orbit:
[1], [2], [3, 4], [5, 6], [7, 10, 8],
[9, 14, 22, 19, 16, 26, 24, 20, 17, 15, 13, 11],
[21, 34, 29, 25],
and the first apparently infinite orbit is, in the forward direction,
[23, 38, 33, 32, 28, 46, 41, 40, 35, 58, 51, 45, 42, 37, 62, 106, ...] (see A282712), and in the reverse direction
[23, 27, 31, 36, 39, 44, 50, 57, 65, 73, 82, 47, 53, 61, 68, 77, ...] (see A282713). (End)
Conjecture: The two lines in the graph are (apart from small local deviations) defined by the same equations as the two lines in the graph of A283312.  N. J. A. Sloane, Mar 12 2017


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..75000 (First 10000 terms from Rémy Sigrist)
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020
N. J. A. Sloane, Proof that A127202 is a permutation.


EXAMPLE

gcd(a(7), a(8)) = gcd(10,7) = 1. So a(9) is the smallest positive integer which does not occur earlier in the sequence and which is such that gcd(a(9), 7) is not 1. So a(9) = 14, since gcd(14,7) = 7.


MATHEMATICA

f[l_List] := Block[{k = 1, c = GCD[l[[ 1]], l[[ 2]]]}, While[MemberQ[l, k]  GCD[k, l[[ 1]]] == c, k++ ]; Append[l, k]]; Nest[f, {1, 2}, 69] (* Ray Chandler, Jan 16 2007 *)


PROG

(PARIGP, based on Rémy Sigrist's program for A280985)
{ seen = 0; p = 1; g = 2;
for (n=1, 10000,
a = 1;
while (bittest(seen, a)  (n>2 && gcd(p, a)==g), a++; );
print (n " " a);
g = gcd(p, a);
p = a;
seen += 2^a;
)
}
quit


CROSSREFS

Cf. A127203, A283312.
Agrees with A280985 for first 719 terms.
For fixed points see A281353. See also A282712, A282713.
Sequence in context: A338221 A283312 A280985 * A179869 A179881 A227113
Adjacent sequences: A127199 A127200 A127201 * A127203 A127204 A127205


KEYWORD

nonn


AUTHOR

Leroy Quet, Jan 08 2007


EXTENSIONS

Extended by Ray Chandler, Jan 16 2007


STATUS

approved



