A291588 - OEIS

%I #20 Jun 07 2024 10:49:59

%S 1,2,3,5,4,7,11,6,13,17,8,19,9,10,23,29,14,27,25,16,31,37,12,35,41,22,

%T 43,39,20,47,49,32,33,53,26,59,61,15,67,71,28,73,45,34,79,77,38,65,83,

%U 46,89,21,40,97,91,44,51,95,58,101,103,18,55,107,52,109

%N Lexicographically earliest sequence of distinct positive terms such that for any n > 0 and k >= 0, gcd(a(n), a(n + 2^k)) = 1.

%C For a nonempty subset of the natural numbers, say S, let f_S be the lexicographically earliest sequence of distinct positive terms such that, for any n > 0 and s in S, gcd(a(n), a(n + s)) = 1:

%C - f_S is well defined (we can always extend the sequence with a new prime number),

%C - f_S(1) = 1, f_S(2) = 2, f_S(3) = 3,

%C - all prime numbers appear in f_S, in increasing order,

%C - if a(k) = p for some prime p, then k <= p and max_{i=1..k} a(i) = p,

%C - in particular:

%C   S               f_S

%C   ---------       ---

%C   { 1 }           A000027 (the natural numbers)

%C   { 2 }           A121216

%C   { 1, 2 }        A084937

%C   { 1, 2, 3 }     A103683

%C   { 1, 2, 3, 4 }  A143345

%C   A000027         A008578 (1 alongside the prime numbers)

%C   A000079         a (this sequence)

%C - see also Links section for the scatterplots of f_S for certain classical S sets,

%C - likely f_S = f_S' iff S = S'.

%C The motivation for this sequence is to have a sequence f_S for some infinite subset S of the natural numbers.

%H Rémy Sigrist, <a href="/A291588/b291588.txt">Table of n, a(n) for n = 1..20000</a>

%H Rémy Sigrist, <a href="/A291588/a291588.png">Scatterplot of the first 5000 terms of f_A000040 (the prime numbers)</a>

%H Rémy Sigrist, <a href="/A291588/a291588_2.png">Scatterplot of the first 5000 terms of f_A000045 (the Fibonacci numbers)</a>

%H Rémy Sigrist, <a href="/A291588/a291588_1.png">Scatterplot of the first 5000 terms of f_A000142 (the factorial numbers)</a>

%H Rémy Sigrist, <a href="/A291588/a291588_4.png">Scatterplot of the first 5000 terms of f_A000244 (the powers of 3)</a>

%H Rémy Sigrist, <a href="/A291588/a291588_3.png">Scatterplot of the first 5000 terms of f_A000312 (A000312(k) = k^k)</a>

%H Rémy Sigrist, <a href="/A291588/a291588.gp.txt">PARI program for A291588</a>

%H Rémy Sigrist, <a href="/A291588/a291588_5.png">Scatterplot of the first 500000 terms</a>

%e a(1) = 1 is suitable.

%e a(2) must be coprime to a(2 - 2^0) = 1.

%e a(2) = 2 is suitable.

%e a(3) must be coprime to a(3 - 2^0) = 2, a(3 - 2^1) = 1.

%e a(3) = 3 is suitable.

%e a(4) must be coprime to a(4 - 2^0) = 3, a(4 - 2^1) = 2.

%e a(4) = 5 is suitable.

%e a(5) must be coprime to a(5 - 2^0) = 5, a(5 - 2^1) = 3, a(5 - 2^2) = 1.

%e a(5) = 4 is suitable.

%e a(6) must be coprime to a(6 - 2^0) = 4, a(6 - 2^1) = 5, a(6 - 2^2) = 2.

%e a(6) = 7 is suitable.

%e a(7) must be coprime to a(7 - 2^0) = 7, a(7 - 2^1) = 4, a(7 - 2^2) = 3.

%e a(7) = 11 is suitable.

%o (PARI) \\ See Links section.

%Y Cf. A000027, A000040, A000045, A000079, A000142, A000244, A000312, A008578, A084937, A103683, A121216, A143345.

%K nonn

%O 1,2

%A _Rémy Sigrist_, Aug 27 2017