A354721 -id:A354721 - cp's OEIS frontend

A354687 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and the difference | a(n) - a(n-1) | is distinct from all previous differences.

1, 2, 4, 8, 14, 6, 3, 12, 22, 10, 5, 20, 34, 16, 32, 52, 13, 26, 48, 15, 36, 9, 33, 44, 18, 46, 23, 69, 21, 28, 58, 24, 56, 7, 42, 78, 27, 72, 30, 55, 11, 66, 104, 38, 19, 76, 116, 29, 87, 141, 39, 91, 35, 85, 17, 102, 40, 100, 25, 90, 153, 45, 114, 50, 120, 192, 51, 68, 142, 54, 130, 208, 60

Offset: 1

Scott R. Shannon, Jun 03 2022

nonn

The terms are concentrated along many different lines, although three lines contain a higher concentration of terms than the others; these are similar to the three lines seen in A064413. See the linked image. The primes do not occur in their natural order, and unlike A064413, the terms proceeding and following a prime term can be high multiples of the prime.
In the first 200000 terms the fixed points are 1,2,6,10,68. It is plausible no more exist although this is unknown. The sequence is conjectured to be a permutation of the positive integers.
See A354721 for the differences between terms.

			a(4) = 8 as a(3) = 4, and 8 is the smallest unused number that shares a factor with 4 and whose difference from the previous term,| 8 - 4 | = 4, has not appeared. Note 6 shares a factor with 4 but | 6 - 4 | = 2, and a difference of 2 has already occurred between as | a(3) - a(2) |, so 6 cannot be chosen.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^14, showing records in red and local minima in blue, highlighting primes in green and fixed points in gold.
Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Cf. A354721, A064413, A354688, A354731, A354087, A352763.

nn = 120; c[] = d[] = 0; a[1] = c[1] = 1; a[2] = c[2] = j = 2; u = 3; Do[Set[k, u]; While[Nand[c[k] == 0, d[Abs[k - j]] == 0, ! CoprimeQ[j, k]], k++]; Set[{a[i], c[k], d[Abs[k - j]]}, {k, i, i}]; j = k; If[k == u, While[c[u] > 0, u++]], {i, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Jun 04 2022 *)

from math import gcd
from sympy import isprime, nextprime
from itertools import count, islice
def agen(): # generator of terms
    aset, diffset, an, mink = {1, 2}, {1}, 2, 3
    yield from [1, 2]
    for n in count(3):
        k = mink
        while k in aset or abs(an-k) in diffset or gcd(an, k) == 1: k += 1
        aset.add(k); diffset.add(abs(k-an)); an = k; yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 73))) # Michael S. Branicky, Jun 04 2022

A354688 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1) and the difference | a(n) - a(n-1) | is distinct from all previous differences.

Original entry on oeis.org

1, 2, 5, 3, 7, 12, 19, 4, 13, 21, 8, 25, 6, 17, 11, 23, 9, 29, 39, 10, 31, 15, 37, 14, 41, 16, 47, 65, 18, 53, 20, 57, 83, 22, 61, 27, 55, 79, 24, 67, 26, 71, 33, 73, 43, 75, 119, 30, 89, 32, 81, 28, 93, 35, 86, 149, 34, 101, 45, 91, 127, 36, 107, 38, 111, 49, 97, 139, 40, 117, 167, 42, 121, 46

Offset: 1

Scott R. Shannon, Jun 03 2022

nonn

All of the terms are concentrated along four lines - this is in contrast to A352588 where they all concentrated along one line. See the linked image. The primes do not occur in their natural order. The sequence is conjectured to be a permutation of the positive integers.

See A354731 for the differences between terms.

			a(6) = 12 as a(5) = 7, and 12 is the smallest unused number that is coprime to 7 and whose difference from the previous term, | 12 - 7 | = 5, has not appeared. Note that 4,6,8,9,10,11 are all coprime to 7 but their differences from 7 have all appeared as differences between previous terms so none can be chosen.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^14, showing records in red and local minima in blue, highlighting primes in green and fixed points in gold.
Scott R. Shannon, Image of the first 10000 terms. The green line is y = n.

Cf. A354731, A352588, A093714, A354687, A354721.

nn = 120; c[] = d[] = 0; a[1] = c[1] = 1; a[2] = c[2] = j = 2; u = 3; Do[Set[k, u]; While[Nand[c[k] == 0, d[Abs[k - j]] == 0, CoprimeQ[j, k]], k++]; Set[{a[i], c[k], d[Abs[k - j]]}, {k, i, i}]; j = k; If[k == u, While[c[u] > 0, u++]], {i, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Jun 04 2022 *)

from math import gcd
from sympy import isprime, nextprime
from itertools import count, islice
def agen(): # generator of terms
    aset, diffset, an, mink = {1}, set(), 1, 2
    yield 1
    for n in count(2):
        k = mink
        while k in aset or abs(an-k) in diffset or gcd(an, k) != 1: k += 1
        aset.add(k); diffset.add(abs(k-an)); an = k; yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 74))) # Michael S. Branicky, Jun 04 2022

A354731 Absolute values of first differences of A354688.

Original entry on oeis.org

1, 3, 2, 4, 5, 7, 15, 9, 8, 13, 17, 19, 11, 6, 12, 14, 20, 10, 29, 21, 16, 22, 23, 27, 25, 31, 18, 47, 35, 33, 37, 26, 61, 39, 34, 28, 24, 55, 43, 41, 45, 38, 40, 30, 32, 44, 89, 59, 57, 49, 53, 65, 58, 51, 63, 115, 67, 56, 46, 36, 91, 71, 69, 73, 62, 48, 42, 99, 77, 50, 125, 79, 75, 83, 85, 81

Offset: 1

Scott R. Shannon, Jun 04 2022

nonn

See A354688 for further details.

			a(3) = 2 as | A354688(4) - A354688(3) | = | 3 - 5 | = 2.

Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^14, showing records in red and local minima in blue, highlighting primes in green and fixed points in gold.
Scott R. Shannon, Image of the first 10000. The green line is y = n.

Cf. A354688, A352588, A093714, A354687, A354721.

nn = 120; c[] = d[] = 0; a[1] = c[1] = 1; a[2] = c[2] = j = 2; u = 3; {1}~Join~Reap[Do[Set[k, u]; While[Nand[c[k] == 0, d[Abs[k - j]] == 0, CoprimeQ[j, k]], k++]; Set[{a[i], c[k], d[Abs[k - j]]}, {k, i, i}]; Sow[Abs[k - j]]; j = k; If[k == u, While[c[u] > 0, u++]], {i, 3, nn}]][[-1, -1]] (* Michael De Vlieger, Jun 04 2022 *)

from math import gcd
from sympy import isprime, nextprime
from itertools import count, islice
def agen(): # generator of terms
    aset, diffset, an, mink = {1}, set(), 1, 2
    for n in count(2):
        k = mink
        while k in aset or abs(an-k) in diffset or gcd(an, k) != 1: k += 1
        aset.add(k); diffset.add(abs(k-an)); yield abs(k-an); an = k
        while mink in aset: mink += 1
print(list(islice(agen(), 76))) # Michael S. Branicky, Jun 04 2022

cp's OEIS Frontend

A354687 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and the difference | a(n) - a(n-1) | is distinct from all previous differences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A354688 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1) and the difference | a(n) - a(n-1) | is distinct from all previous differences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A354731 Absolute values of first differences of A354688.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python