A291603 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, a(n) is coprime to a(2*n) and to a(2*n+1).

1, 2, 3, 5, 7, 4, 8, 6, 9, 10, 11, 13, 15, 17, 19, 23, 25, 14, 16, 21, 27, 12, 18, 20, 22, 26, 28, 24, 29, 30, 31, 32, 33, 34, 36, 37, 39, 35, 41, 38, 40, 43, 44, 47, 49, 53, 55, 51, 57, 45, 59, 61, 63, 65, 67, 71, 73, 42, 46, 77, 79, 48, 50, 69, 75, 52, 56

Offset

1,2

Comments

This sequence has connections with A082746: here a(n) is coprime to a(2*n) and to a(2*n+1), there a(n) is coprime to a(2*n).

This sequence is a permutation of the natural numbers (with inverse A291604 and fixed points A291610):

- the sequence can always be extended with a prime number,

- all prime numbers appear in the sequence, in increasing order,

- for any n, there are infinitely many prime numbers coprime to n, so eventually n will appear in the sequence.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing records in red and local minima in blue, highlighting primes in green and other prime powers in gold.

Rémy Sigrist, Scatterplot of the first 10000 terms

Rémy Sigrist, PARI program for A291603

Index entries for sequences that are permutations of the natural numbers

Example

a(1) = 1 is suitable.
a(2) must be coprime to a(1) = 1.
a(2) = 2 is suitable.
a(3) must be coprime to a(1) = 1.
a(3) = 3 is suitable.
a(4) must be coprime to a(2) = 2.
a(4) = 5 is suitable.
a(5) must be coprime to a(2) = 2.
a(5) = 7 is suitable.
a(6) must be coprime to a(3) = 3.
a(6) = 4 is suitable.
a(7) must be coprime to a(3) = 3.
a(7) = 8 is suitable.
a(8) must be coprime to a(4) = 5.
a(8) = 6 is suitable.
a(9) must be coprime to a(4) = 5.
a(9) = 9 is suitable.
a(10) must be coprime to a(5) = 7.
a(10) = 10 is suitable.

Mathematica

nn = 67; c[_] = False; Set[{a[1], c[1]}, {1, True}]; u = 2; Do[Set[{j, k}, {a[Floor[n/2]], u}]; While[Nand[! c[k], CoprimeQ[j, k]], k++]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Oct 28 2022 *)

Prog

See Links section.

Crossrefs

Cf. A082746, A291604 (inverse), A291610 (fixed points).

Sequence in context: A082196 A292876 A126049 * A082331 A140528 A065037

Adjacent sequences: A291600 A291601 A291602 * A291604 A291605 A291606

Keyword

nonn,look

Author

Rémy Sigrist, Aug 27 2017

Status

approved