A121216 -id:A121216 - cp's OEIS frontend

A340807 a(n) = n if n <= 3; for n > 3, a(n) is the closest number to a(n-2) that has not occurred earlier and has at least one common factor with a(n-2), but none with a(n-1). In case of a tie, choose the smaller.

1, 2, 3, 4, 9, 8, 15, 14, 5, 12, 25, 16, 35, 18, 49, 20, 63, 22, 57, 26, 51, 28, 45, 32, 39, 34, 33, 38, 27, 40, 21, 44, 7, 46, 77, 48, 91, 50, 117, 55, 114, 65, 112, 75, 116, 69, 118, 81, 122, 87, 124, 99, 128, 93, 130, 111, 134, 105, 136, 95, 138, 85, 141, 80, 147, 82, 153, 86, 159, 88

Offset: 1

Scott R. Shannon, Jan 22 2021

The sequence uses a similar selection rule to the Yellowstone permutation A098550 but instead of choosing the smallest number that has not occurred earlier that has a common factor with a(n-2) and no common factor with a(n-1), the number closest to a(n-2) that satisfies these rules is selected for a(n). If two such numbers are the same distance from a(n-2) then the smaller is chosen.
Many terms are clustered along a line with gradient approximately 1.33. However, along this line the terms often rapidly drop to much smaller values before returning to the main line. More interesting is the existence of regions on the same line where the terms split and form two lines of constantly increasing values. These lines continue until they both start decreasing again to rejoin near the original line.
In the first 15 million terms the maximum number of consecutive increasing terms is seven. This run starts at n = 47685. The maximum number of consecutive decreasing terms is also seven. This starts at n = 4134621.
In the first 15 million terms the fixed points, other than the first three terms, are 4, 323, 516718, 2199679, 2401224. As the terms for larger n seem to drop below the a(n)=n line on numerous occasions, it is possible that more exist, although this is unknown. The smallest number not appearing is 6, although other small values appear after many terms, e.g. a(4946191) = 23. It is unknown if all values eventually appear. The largest change in consecutive terms is from a(399922)=527754 to a(399923)=2887, a difference of 524867.
See also A340783 where the next term is the closest to a(n-1).

			a(5) = 9 as a(5-2) = a(3) = 3 so a(5) must have 3 as a factor, but cannot be 6 = 3*2 as it cannot have a common factor with a(5-1) = a(4) = 2.
a(12) = 16 as a(12-2) = a(10) = 12 so a(12) must have 2 or 3 as a factor, but cannot have a common factor with a(12-1) = a(11) = 25 = 5*5. The closest numbers to a(12-2) = a(10) = 12 which have 2 or 3 as a factor but not 5 are 8,9,14,16. The first three have already appeared so a(12) = 16.

Scott R. Shannon, Table of n, a(n) for n = 1..10000.
Scott R. Shannon, Image of the first 500 thousand terms. The green line is a(n)=n.
Scott R. Shannon, Image of the first 15 million terms. The green line is a(n)=n.
Rémy Sigrist, PARI program for A340807

Cf. A098550, A340783, A336957, A064413, A121216.

PARI
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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.

Original entry on oeis.org

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

Rémy Sigrist, Aug 27 2017

nonn

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.

			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.

Rémy Sigrist, Table of n, a(n) for n = 1..20000
Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000040 (the prime numbers)
Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000045 (the Fibonacci numbers)
Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000142 (the factorial numbers)
Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000244 (the powers of 3)
Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000312 (A000312(k) = k^k)
Rémy Sigrist, PARI program for A291588
Rémy Sigrist, Scatterplot of the first 500000 terms

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

PARI
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A344307 a(n) is the smallest number not yet in the sequence that satisfies the following condition: if p is a prime factor of a(n), the next p terms are coprime to p.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 8, 11, 13, 6, 17, 19, 10, 21, 23, 16, 29, 27, 20, 31, 37, 12, 41, 43, 14, 15, 47, 22, 53, 39, 32, 25, 49, 18, 59, 61, 34, 45, 67, 38, 71, 33, 28, 65, 73, 24, 79, 83, 46, 75, 89, 56, 97, 81, 44, 85, 101, 26, 87, 103, 62, 35, 57, 64, 107

Offset: 1

Sergio Pimentel, May 14 2021

nonn

It is not clear if every positive integer is in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A344307

Cf. A064413, A084937, A090252, A103683, A121216, A249064.

PARI
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See Links section.
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More terms from Rémy Sigrist, May 29 2021

A340779 a(1)=1, a(2)=2; for n>=3, a(n) = the closest number to a(n-1) that has not occurred earlier and has at least one common factor with a(n-1), but none with a(n-2). In case of a tie, choose the smaller.

Original entry on oeis.org

1, 2, 6, 15, 35, 28, 26, 39, 33, 22, 20, 45, 51, 34, 38, 57, 63, 56, 58, 87, 93, 62, 68, 85, 75, 72, 74, 185, 175, 168, 166, 415, 405, 402, 404, 505, 495, 492, 494, 481, 407, 396, 394, 985, 975, 972, 974, 2435, 2425, 2328, 2326, 5815, 5805, 5802, 5804, 7255, 7245, 7242, 7244, 9055

Offset: 1

Scott R. Shannon, Jan 21 2021

nonn

The sequence uses a similar selection rule to the Enots Wolley sequence A336957 but instead of choosing the smallest number that has not occurred earlier that has a common factor with a(n-1) and no common factor with a(n-2), the number closest to a(n-1) that satisfies these rules is selected for a(n). If two such numbers are the same distance from a(n-1) then the smaller is chosen. Like A336957 for the sequence to continue a(n) must always have a prime factor not in a(n-1), thus a(n) cannot be a prime or a prime power.

The sequence grows sporadically with n, showing regions of little growth followed by a large jump due to the next term being the multiple of a large prime of the previous term. However due to the overall rapid increase in the terms it is very unlikely any fixed points exist.

			a(5) = 35 as a(4) = 15 = 3*5 and a(3) = 6 = 2*3, thus a(5) must be a multiple of 5 while not being a multiple of 3, and must have a prime factor other than 5. The smallest unused number closest to 15 satisfying these criteria is 35.
a(6) = 28 as a(5) = 35 = 5*7 and a(4) = 15 = 3*5, this a(6) must be a multiple of 7 while not being a multiple of 5, and must have a prime factor other than 7. The smallest number satisfying these criteria is 14. However 28 also does and is only 7 away from a(5), while 14 is 21 away, thus 28 is chosen. This is the first term that differs from A336957.

Cf. A336957, A098550, A340783, A340807, A064413, A121216.

cp's OEIS Frontend

A340807 a(n) = n if n <= 3; for n > 3, a(n) is the closest number to a(n-2) that has not occurred earlier and has at least one common factor with a(n-2), but none with a(n-1). In case of a tie, choose the smaller.

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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.

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A344307 a(n) is the smallest number not yet in the sequence that satisfies the following condition: if p is a prime factor of a(n), the next p terms are coprime to p.

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A340779 a(1)=1, a(2)=2; for n>=3, a(n) = the closest number to a(n-1) that has not occurred earlier and has at least one common factor with a(n-1), but none with a(n-2). In case of a tie, choose the smaller.

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