A354688 -id:A354688 - cp's OEIS frontend

A354731 Absolute values of first differences of A354688.

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

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.

Original entry on oeis.org

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

A354727 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.

Original entry on oeis.org

1, 2, 4, 8, 6, 3, 9, 12, 20, 5, 10, 22, 11, 33, 15, 24, 14, 7, 21, 28, 16, 26, 13, 39, 18, 34, 17, 51, 27, 42, 36, 32, 50, 25, 45, 40, 64, 30, 55, 35, 56, 48, 75, 66, 38, 19, 57, 76, 44, 72, 46, 23, 69, 92, 52, 65, 95, 60, 93, 31, 62, 94, 47, 141, 54, 90, 63, 49, 84, 68, 85, 125, 70, 112, 58

Offset: 1

Scott R. Shannon, Jun 05 2022

nonn

This sequences uses similar a similar rule to A354687 but here the sign of the difference between a(n-1) and a(n) is considered. The terms show an overall similar behavior to A354687 although here only two lines show a higher concentration of terms than the other lines. See the linked image.
In the first 100000 terms the fixed points are 1,2,15,32,100,115,300,720. It is plausible no more exist although this is unknown.. The sequence is conjectured to be a permutation of the positive integers.
See A354739 for the differences between terms.

			a(9) = 20 as a(8) = 12, and 20 is the smallest unused number that shares a factor with 12 and whose difference from the previous term, 20 - 12 = 8, has not appeared. Note that 10,14,15,16,18 all share a factor with 12 but their differences from 12, namely -2,2,3,4,6, have already appeared as differences between previous pairs of terms.

Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Cf. A354739, A354687, A354575, A354679, A354688.

A354575 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, 4, 9, 8, 15, 11, 6, 17, 10, 19, 13, 21, 23, 12, 25, 16, 31, 14, 33, 20, 37, 18, 41, 26, 47, 22, 49, 27, 43, 29, 35, 53, 24, 55, 28, 57, 34, 59, 38, 71, 30, 67, 32, 73, 36, 79, 39, 61, 45, 77, 46, 81, 91, 40, 87, 44, 83, 50, 99, 52, 97, 42, 95, 51, 65, 89, 63, 101, 48, 103, 54, 113

Offset: 1

Scott R. Shannon, Jun 05 2022

This sequence uses a similar rule to A354688 but here the sign of the difference between a(n-1) and a(n) is considered. This leads to the terms showing much more erratic behavior than A354688; see the linked image.
In the first 200000 terms the fixed points are 1,2,8,35, and it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.
See A354679 for the differences between terms.

			a(9) = 15 as a(8) = 8, and 15 is the smallest unused number that is coprime to 8 and whose difference from the previous term, 15 - 8 = 7, has not appeared. Note that 11 and 13 are coprime to 8 but their differences from 8, namely 3 and 5, have already appeared as differences between previous pairs of terms.
a(15) = 13 as a(14) = 19, and 13 is the smallest unused number that is coprime to 19 and whose difference from the previous term, 13 - 19 = -6, has not appeared. Note that 12 is coprime to 19 and smaller than 13 but its difference from 19, namely -7, has already appeared as a difference between a(13) and a(12).

Scott R. Shannon, Image of the first 200000 terms. The green line is y = n.

Cf. A354679, A354688, A354727, A354739, A354687.

A354679 First differences of A354575.

Original entry on oeis.org

1, 3, -2, 4, -3, 5, -1, 7, -4, -5, 11, -7, 9, -6, 8, 2, -11, 13, -9, 15, -17, 19, -13, 17, -19, 23, -15, 21, -25, 27, -22, 16, -14, 6, 18, -29, 31, -27, 29, -23, 25, -21, 33, -41, 37, -35, 41, -37, 43, -40, 22, -16, 32, -31, 35, 10, -51, 47, -43, 39, -33, 49, -47, 45, -55, 53, -44, 14, 24, -26

Offset: 1

Scott R. Shannon, Jun 05 2022

See A354575 for further details.

			a(3) = -2 as A354575(4) - A354575(3) = 3 - 5 = -2.

Scott R. Shannon, Image of the first 200000 terms. The blue line is y = 0.

Cf. A354575, A354688, A354727, A354739, A354687.

A354721 Absolute values of first differences of A354687.

Original entry on oeis.org

1, 2, 4, 6, 8, 3, 9, 10, 12, 5, 15, 14, 18, 16, 20, 39, 13, 22, 33, 21, 27, 24, 11, 26, 28, 23, 46, 48, 7, 30, 34, 32, 49, 35, 36, 51, 45, 42, 25, 44, 55, 38, 66, 19, 57, 40, 87, 58, 54, 102, 52, 56, 50, 68, 85, 62, 60, 75, 65, 63, 108, 69, 64, 70, 72, 141, 17, 74, 88, 76, 78, 148, 80, 91, 77, 81

Offset: 1

Scott R. Shannon, Jun 04 2022

nonn

See A354687 for further details.

			a(5) = 8 as | A354687(6) - A354687(5) | = | 6 - 14 | = 8.

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. A354687, A064413, A354688, A354731, A354087, A352763.

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, 2}, {1}, 2, 3
    yield from [1]
    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)); yield abs(an-k); an = k
        while mink in aset: mink += 1
print(list(islice(agen(), 76))) # Michael S. Branicky, Jun 04 2022

A354739 First differences of A354727.

Original entry on oeis.org

1, 2, 4, -2, -3, 6, 3, 8, -15, 5, 12, -11, 22, -18, 9, -10, -7, 14, 7, -12, 10, -13, 26, -21, 16, -17, 34, -24, 15, -6, -4, 18, -25, 20, -5, 24, -34, 25, -20, 21, -8, 27, -9, -28, -19, 38, 19, -32, 28, -26, -23, 46, 23, -40, 13, 30, -35, 33, -62, 31, 32, -47, 94, -87, 36, -27, -14, 35, -16, 17

Offset: 1

Scott R. Shannon, Jun 05 2022

sign

See A354727 for further details.

			a(5) = -3 as A354727(6) - A354727(5) = 3 - 6 = -3.

Scott R. Shannon, Image of the first 100000 terms. The blue line is y = 0.

Cf. A354727, A354687, A354575, A354679, A354688.

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A354731 Absolute values of first differences of A354688.

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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.

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A354727 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.

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A354575 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.

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A354679 First differences of A354575.

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A354721 Absolute values of first differences of A354687.

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A354739 First differences of A354727.

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