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A291588
Lexicographically earliest sequence of distinct positive terms such that for any n > 0 and k >= 0, gcd(a(n), a(n + 2^k)) = 1.
1
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, 43, 39, 20, 47, 49, 32, 33, 53, 26, 59, 61, 15, 67, 71, 28, 73, 45, 34, 79, 77, 38, 65, 83, 46, 89, 21, 40, 97, 91, 44, 51, 95, 58, 101, 103, 18, 55, 107, 52, 109
OFFSET
1,2
COMMENTS
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:
- f_S is well defined (we can always extend the sequence with a new prime number),
- f_S(1) = 1, f_S(2) = 2, f_S(3) = 3,
- all prime numbers appear in f_S, in increasing order,
- if a(k) = p for some prime p, then k <= p and max_{i=1..k} a(i) = p,
- in particular:
S f_S
--------- ---
{ 1 } A000027 (the natural numbers)
{ 2 } A121216
{ 1, 2 } A084937
{ 1, 2, 3 } A103683
{ 1, 2, 3, 4 } A143345
A000027 A008578 (1 alongside the prime numbers)
A000079 a (this sequence)
- see also Links section for the scatterplots of f_S for certain classical S sets,
- likely f_S = f_S' iff S = S'.
The motivation for this sequence is to have a sequence f_S for some infinite subset S of the natural numbers.
EXAMPLE
a(1) = 1 is suitable.
a(2) must be coprime to a(2 - 2^0) = 1.
a(2) = 2 is suitable.
a(3) must be coprime to a(3 - 2^0) = 2, a(3 - 2^1) = 1.
a(3) = 3 is suitable.
a(4) must be coprime to a(4 - 2^0) = 3, a(4 - 2^1) = 2.
a(4) = 5 is suitable.
a(5) must be coprime to a(5 - 2^0) = 5, a(5 - 2^1) = 3, a(5 - 2^2) = 1.
a(5) = 4 is suitable.
a(6) must be coprime to a(6 - 2^0) = 4, a(6 - 2^1) = 5, a(6 - 2^2) = 2.
a(6) = 7 is suitable.
a(7) must be coprime to a(7 - 2^0) = 7, a(7 - 2^1) = 4, a(7 - 2^2) = 3.
a(7) = 11 is suitable.
PROG
(PARI) \\ See Links section.
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Aug 27 2017
STATUS
approved