A084937 -id:A084937 - cp's OEIS frontend

A180348 Lexicographically earliest sequence such that a(n) is coprime to the preceding 4 terms and does not occur earlier; a(n) = n for n=1,2,3,4.

1, 2, 3, 4, 5, 7, 11, 9, 8, 13, 17, 19, 15, 14, 23, 29, 31, 25, 6, 37, 41, 43, 35, 12, 47, 53, 59, 49, 10, 27, 61, 67, 71, 16, 21, 55, 73, 79, 26, 51, 77, 83, 89, 20, 39, 97, 101, 103, 22, 45, 91, 107, 109, 32, 33, 65, 113, 119, 38, 69, 121, 125, 127, 28, 57, 131, 85, 137, 44, 63, 139, 95, 149, 34, 81, 143, 115, 133, 58

Offset: 1

M. F. Hasler, Jan 18 2011

nonn

One possible generalization of A084937 resp. A103683 to N=4, cf. A143345 for a different version.

N=99; print1("1, 2, 3"); a=[1, 2, 3, L=4]; unused=[]; v=vector(#a, i, 1); for(n=#a, N, print1(", "a[#a]); for(i=1, #unused, apply(x->gcd(x, unused[i]), a)==v || next; a=concat(vecextract(a, "^1"), unused[i]); unused=vecextract(unused, Str("^", i)); next(2)); L++; while(apply(x->gcd(x, L), a) !=v, unused=concat(unused, L++-1); ); a=concat(vecextract(a, "^1"), L))

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
```
\\ See Links section.
```

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
```
See Links section.
```

More terms from Rémy Sigrist, May 29 2021

A115928 Smallest number which is coprime to the last four predecessors and not occurring earlier; a(1)=1, a(2)=2, a(3)=3 & a(4)=4.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 8, 11, 13, 15, 14, 17, 19, 23, 6, 25, 29, 31, 12, 35, 37, 41, 16, 21, 43, 47, 10, 27, 49, 53, 20, 33, 59, 61, 26, 45, 67, 71, 22, 39, 73, 79, 28, 51, 55, 83, 32, 57, 65, 77, 34, 69, 89, 91, 38, 75, 97, 101, 44, 63, 85, 103, 46, 81, 195, 107, 52, 87, 109, 113, 40, 93, 119

Offset: 1

Robert G. Wilson v, Jun 26 2011

Cf. A084937, A103683.

f[s_] := Block[{k = 1, l = Take[s,-4]}, While[ Union[ GCD[k, l]] != {1} || MemberQ[s,k], k++]; Append[s,k]]; Nest[f, {1,2,3,4}, 70]

cp's OEIS Frontend

A180348 Lexicographically earliest sequence such that a(n) is coprime to the preceding 4 terms and does not occur earlier; a(n) = n for n=1,2,3,4.

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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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Extensions

A115928 Smallest number which is coprime to the last four predecessors and not occurring earlier; a(1)=1, a(2)=2, a(3)=3 & a(4)=4.

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