A093715 -id:A093715 - cp's OEIS frontend

A093714 a(n) = smallest number coprime to a(n-1), not equal to a(n-1)+1, and not occurring earlier; a(1)=1.

1, 3, 2, 5, 4, 7, 6, 11, 8, 13, 9, 14, 17, 10, 19, 12, 23, 15, 22, 21, 16, 25, 18, 29, 20, 27, 26, 31, 24, 35, 32, 37, 28, 33, 38, 41, 30, 43, 34, 39, 44, 47, 36, 49, 40, 51, 46, 45, 52, 55, 42, 53, 48, 59, 50, 57, 56, 61, 54, 65, 58, 63, 62, 67, 60, 71, 64, 69, 68, 73, 66, 79

Offset: 1

Reinhard Zumkeller, Apr 12 2004

nonn

Lexicographically earliest infinite sequence of distinct positive numbers such that gcd(a(n-1), a(n)) = 1, a(n) != a(n-1) + 1. - N. J. A. Sloane, May 02 2022
Permutation of natural numbers with inverse A093715: a(A093715(n))=A093715(a(n))=n.
Comments from N. J. A. Sloane, May 02 2022: (Start)
Proof that this is a permutation of the natural numbers.
1. As usual for a "lexicographically earliest sequence" of this class, there is a function B(k) such that a(n) > k for all n > B(k).
2. For any prime p, p divides a(n) for some n. [Suppose not. Using 1, find n_0 such that a(n) > p^2 for all n >= n_0. But if a(n) > p^2, then p is a smaller choice for a(n+1), contradiction.]
3. For any prime p, p divides infinitely many terms. [Suppose not. Let p^i be greater than any multiple of p in the sequence. Go out a long way, and find a term greater than p^i. Then p^i is a smaller candidate for the next term. Contradiction.]
4. Every prime p appears naked. [If not, using 3, find a large multiple of p, G*p, say. But then p would have been a smaller candidate than G*p. Contradiction.]
5. The next term after a prime p is the smallest number not yet in the sequence which is relatively prime to p. Suppose k is missing from the sequence, and find a large prime p that does not divide k. Then the term after p will be k.  So every number appears.
This completes the proof.
Conjecture 1: If p is a prime >= 3, p-1 appears after p.
Conjecture 2: If p is a prime, the first term divisible by p is p itself.
Conjecture 3: If a(n) = p is a prime >= 5, then n < p.
(End)
Coincides with A352588 for n >= 17. - Scott R. Shannon, May 02 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A085229, A347113, A352588, A352928 (smallest missing number).
A352929 gives indices of prime terms, A352930 = first differences, A352931 = a(n)-n. See also A352932.
See Comments in A109812 for a set-theory analog.

nn = 120; c[] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, CoprimeQ[#, k], k != # + 1], k++] &@ a[i - 1]; Set[{a[i], c[k]}, {k, i}]; If[a[i] == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, May 02 2022 *)

from math import gcd
from itertools import islice
def agen(): # generator of terms
    an, aset, mink = 1, {1}, 2
    while True:
        yield an
        k = mink
        while k in aset or gcd(an, k) != 1 or k-an == 1: k += 1
        an = k
        aset.add(an)
        while mink in aset: mink += 1
print(list(islice(agen(), 72))) # Michael S. Branicky, May 02 2022

A085230 Inverse of A085229.

Original entry on oeis.org

1, 3, 2, 5, 4, 7, 6, 9, 10, 11, 8, 13, 12, 15, 16, 17, 14, 19, 18, 21, 22, 23, 20, 25, 24, 27, 28, 29, 26, 31, 30, 33, 34, 35, 36, 37, 32, 39, 40, 41, 38, 43, 42, 45, 46, 47, 44, 49, 48, 51, 52, 53, 50, 55, 54, 57, 58, 59, 56, 61, 60, 63, 64, 65, 66, 67, 62, 69, 70

Offset: 1

Reinhard Zumkeller, Jun 22 2003

nonn

Permutation of natural numbers: a(A085229(n)) = A085229(a(n)) = n.

Odd numbers appear in order, at a(2k) for k > 0. - Michael De Vlieger, Apr 13 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A085229, A093715.

nn = 10000; c[] = 0; a[1] = c[1] = 1; a[2] = 3; c[3] = u = 2; Do[k = u; While[Nand[c[k] == 0, CoprimeQ[a[i-1], k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[a[i] == u, While[c[u] > 0, u++]], {i, 3, nn}]; Array[c, nn] (* _Michael De Vlieger, Apr 13 2022 *)

A353706 n appears in A093714 a(n) steps earlier than expected.

Original entry on oeis.org

0, -1, 1, -1, 1, -1, 1, -1, -2, -4, 3, -4, 3, 2, -3, -5, 4, -5, 4, -5, 1, 3, 6, -5, 3, -1, 1, -5, 5, -7, 3, 1, -1, -5, 5, -7, 5, 3, -1, -5, 5, -9, 5, 3, -3, -1, 5, -5, 5, -5, 5, 3, 1, -5, 5, -1, 1, -3, 5, -5, 3, -1, 1, -3, 5, -5, 3, -1, 1, -3, 5, -5

Offset: 1

N. J. A. Sloane, May 05 2022

sign

It appears that a positive number n appears in A093714 at about the n-th term; if in fact A093714(k) = n, then a(n) = n - k.

An equivalent definition uses the inverse to A093714: a(n) = n - A093715(n).

			A093714(8) = 11, so 11 has appeared 3 steps early, therefore a(11) = 11 - 8 = 3.

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

Cf. A093714, A093715, A353707.

PARI
```
See Links section.
```

cp's OEIS Frontend

A093714 a(n) = smallest number coprime to a(n-1), not equal to a(n-1)+1, and not occurring earlier; a(1)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

A085230 Inverse of A085229.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A353706 n appears in A093714 a(n) steps earlier than expected.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI