A352588 -id:A352588 - 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

A351495 a(1) = 1, for n > 1, a(n) is the smallest positive number that has not yet appeared that is a multiple of the smallest prime that does not divide a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 9, 10, 12, 15, 14, 18, 20, 21, 16, 24, 25, 22, 27, 26, 30, 7, 28, 33, 32, 36, 35, 34, 39, 38, 42, 40, 45, 44, 48, 50, 51, 46, 54, 55, 52, 57, 56, 60, 49, 58, 63, 62, 66, 65, 64, 69, 68, 72, 70, 75, 74, 78, 80, 81, 76, 84, 85, 82, 87, 86, 90, 77, 88, 93, 92, 96, 95, 94

Offset: 1

Scott R. Shannon, May 03 2022

nonn

The sequence is conjectured to be a permutation of the positive integers.

The k-th prime appears as the next term after A002110(k-1) appears.

			a(5) = 6 as a(4) = 2 = 2*2 which does not contain 3 as a prime factor, and 6 is the smallest unused number that is a multiple of 3.
a(6) = 5 as a(5) = 6 = 2*3 which does not contain 5 as a prime factor, and 5 is the smallest unused number that is a multiple of 5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16, showing m with lpf(m) = 7 in red, lpf(m) = 11 in gold, lpf(m) = 13 in green, and lpf(m) = 17 in blue.
Scott R. Shannon, Image of the first 50000 terms. The green line is y = n.

Cf. A353026, A352793, A002110, A000040, A064413, A352588, A359804,

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

A352932 Where the parity of A352931 changes.

Original entry on oeis.org

1, 2, 11, 18, 79, 1788, 15023, 38124, 45053, 1036044, 100280245077

Offset: 1

N. J. A. Sloane, May 04 2022

The terms 15023, 38124, 45053, 1036044 are based on the comments from Scott R. Shannon and Michael De Vlieger in A352588.

In fact a(n) = A351498(n) + 1, although this requires proof (see A351498).

Rémy Sigrist, C program

Cf. A093714, A351498, A352588, A352931.

C
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See Links section.
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a(9) corrected and a(11) added by Rémy Sigrist, May 06 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

A372975 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) shares a factor with a(n-1) while omega(a(n)) does not equal omega(a(n-1)).

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 10, 5, 15, 9, 18, 8, 14, 7, 21, 27, 24, 16, 20, 25, 30, 22, 11, 33, 42, 26, 13, 39, 60, 28, 32, 34, 17, 51, 66, 36, 64, 38, 19, 57, 78, 40, 70, 35, 49, 56, 84, 44, 90, 45, 81, 48, 102, 46, 23, 69, 105, 50, 110, 52, 114, 54, 120, 55, 121, 77, 126, 58, 29, 87, 132, 62, 31, 93, 138

Offset: 1

Scott R. Shannon, May 26 2024

nonn

The sequence shows similar behavior to the EKG sequence A064413; for the terms studied the primes appear in the natural order, and when a prime p is a term, the proceeding and following terms are 2p and 3p respectively.
For larger n a graph of the sequence also displays very similar behavior to A064413, although for the first ~2500 terms the main concentration of terms are along two lines which eventually join - see the attached image of the first 5000 terms.
The fixed points begin 1, 2, 22, 26, 36, 38, 1991, 2023, 2159, 2189, 2627; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.
From Michael De Vlieger, May 28 2024: (Start)
Four general trajectories become apparent in log log scatterplot:
1. Beta, the trajectory of primes a(j) = p.
2. Alpha, the trajectory of numbers a(j+1) = 3*p.
3. Delta, the trajectory of perfect prime powers and numbers k with omega(k) = 3.
4. Gamma, the trajectory of all other (composite) numbers.
Delta begins with a(21) = 30 and merges with gamma around n = 2958. The merger alters the "slope" of all trajectories as a result. Thereafter, a number k with omega(k) = 2 is comparable in size with one that has omega(k) = 3. This does not seem to happen for omega(k) = 4, etc. (See a(2102) = 2310).
Perfect prime powers may technically constitute a separate, more scattered trajectory superposed upon delta. Still, the merger with gamma seems to occur around the same point as with delta.
Exception to first comment: 12 follows 3, since omega(9) = omega(3). The number 12 lies outside trajectory alpha, since 12 = 4*3. (End)

			a(3) = 6 as a(2) = 2 and omega(2) = A001221(2) = 1, and 6 shares a factor with 2 while omega(6) = A001221(6) = 2 which does not equal 1.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 5000 terms. The green line is a(n) = n.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple represents powerful numbers that are not prime powers.

Cf. A001221, A027748, A372974, A064413, A352588, A109465, A353916, A372974.

nn = 1000; c[] := False; m[] := 1;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = 2; v = 4;
Do[Which[
  And[PrimeQ[#], OddQ[#]] &[j/2], k = j/2,
  PrimePowerQ[j], k = FactorInteger[j][[1, 1]];
    While[Or[c[#], PrimePowerQ[#]] &[k*m[k]], m[k]++]; k *= m[k],
  True, k = v;
    While[Or[CoprimeQ[j, k], PrimeNu[k] == #, c[k]] &[PrimeNu[j]], k++]];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == v, While[Or[PrimeQ[v], c[v]], v++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, May 28 2024 *)

A372974 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) is coprime to a(n-1) and omega(a(n)) does not equal omega(a(n-1)).

Original entry on oeis.org

1, 2, 15, 4, 21, 5, 6, 7, 10, 3, 14, 9, 20, 11, 12, 13, 18, 17, 22, 19, 24, 23, 26, 25, 28, 27, 34, 29, 30, 31, 33, 8, 35, 16, 39, 32, 45, 37, 36, 41, 38, 43, 40, 47, 42, 53, 44, 49, 46, 59, 48, 61, 50, 67, 51, 64, 55, 71, 52, 73, 54, 79, 56, 81, 58, 83, 57, 70, 69, 89, 60, 77, 78, 85, 66, 65, 84

Offset: 1

Scott R. Shannon, May 26 2024

nonn

The fixed points show an unusual pattern; they begin 1, 2, 4, 69, 190, 438, 545, 725, 732, 909 and it appears, based on a graph of the sequence (see the attached image of the first 5000 terms) there may be no more. However more exist at 324388, 330574, 333069, 333531,..., 369752. Then once again there is a large gap until 2704713, 2726054, 2760963, ... . It is unclear what causes this behavior.

The sequence is conjectured to be a permutation of the positive integers.

			a(3) = 15 as a(2) = 2 and omega(2) = A001221(2) = 1, and 15 is coprime to 2 while omega(15) = A001221(15) = 2 which does not equal 1. No smaller number satisfies both of these requirements.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 5000 terms. The green line is a(n) = n.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple, with purple indicating powerful numbers that are not prime powers.

Cf. A001221, A027748, A372975, A352588, A109465.

nn = 120; c[_] := False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = 2; u = 3;
Do[k = u;
  While[Or[GCD[j, k] > 1, PrimeNu[k] == #, c[k]] &[PrimeNu[j]], k++];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, May 28 2024 *)

A370500 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) does not share a factor with a(n-1) but sopfr(a(n)) does share a factor with soprf(a(n-1)), where sopfr(k) is the sum of the primes dividing k, with repetition.

Original entry on oeis.org

1, 2, 9, 4, 15, 8, 3, 14, 27, 20, 77, 16, 21, 5, 6, 25, 18, 35, 24, 65, 32, 33, 7, 10, 49, 12, 115, 36, 55, 39, 50, 51, 26, 81, 38, 105, 44, 91, 30, 119, 57, 11, 28, 45, 62, 85, 42, 95, 64, 69, 13, 22, 63, 74, 75, 56, 169, 60, 121, 40, 123, 70, 87, 98, 93, 17, 52, 99, 145, 66, 133, 72, 125, 46

Offset: 1

Scott R. Shannon, Feb 20 2024

nonn

The fixed points begin 1, 2, 4, 56, 72, 138, 200, 438, 500, 540, 590, 3998. The sequence is conjectured to be a permutation of the positive integers.

			a(4) = 4 as a(3) = 9 and 4 does not share a factor with 9 while sopfr(4) = 4 does share a factor with sopfr(9) = 6.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A001414, A370501, A370047, A370496, A352588.

A370501 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) shares a factor with a(n-1) but sopfr(a(n)) does not share a factor with soprf(a(n-1)), where sopfr(k) is the sum of the primes dividing k, with repetition.

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 10, 5, 15, 20, 16, 14, 7, 21, 24, 18, 22, 8, 28, 26, 13, 39, 27, 30, 34, 17, 51, 45, 9, 48, 32, 38, 19, 57, 63, 33, 11, 44, 40, 25, 75, 35, 56, 36, 52, 42, 46, 23, 69, 54, 50, 58, 29, 87, 90, 55, 80, 60, 76, 62, 31, 93, 96, 64, 82, 41, 123, 99, 66, 68, 85, 105, 49, 112

Offset: 1

Scott R. Shannon, Feb 20 2024

nonn

The fixed points begin 1, 2, 64, 114, 116, 132, and it is plausible no more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(6) = 4 as a(5) = 12 and 4 shares a factor with 12 while sopfr(4) = 4 does not share a factor with sopfr(12) = 7.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A001414, A370500, A370047, A370496, A352588.

A351624 Fixed points of A093714.

Original entry on oeis.org

1, 91, 93, 117, 147, 164, 172, 189, 224, 231, 254, 273, 284, 327, 344, 357, 374, 382, 399, 411, 434, 464, 483, 494, 543, 561, 567, 584, 592, 609, 644, 674, 693, 704, 747, 777, 794, 819, 854, 884, 891, 903, 914, 932, 939, 957, 987, 1001, 1012, 1029, 1064, 1094

Offset: 1

Rémy Sigrist, May 04 2022

nonn

From Michael De Vlieger, May 04 2022: (Start)
Also positions of 0's in A352931.
Fixed points of A352588 are these terms together with 2 and 8. (End)

			A093714(91) = 91 so 91 belongs to this sequence.
A093714(92) = 88 so neither 88 nor 92 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Logarithmic scatterplot of the first differences of the first 400000 terms
Rémy Sigrist, C program

Cf. A093714, A352588, A352931.

C
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See Links section.
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{1}~Join~Reap[Do[k = u; While[Nand[FreeQ[c, k], CoprimeQ[j, k], k != j + 1], k++]; j = k; AppendTo[c, k]; If[k == i, Sow[k]]; If[k == u, While[MemberQ[c, u], u++]; c = DeleteCases[c, ?(# < u &)]], {i, 2, nn}]][[-1, -1]] (* _Michael De Vlieger, May 04 2022 *)

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.

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A351495 a(1) = 1, for n > 1, a(n) is the smallest positive number that has not yet appeared that is a multiple of the smallest prime that does not divide a(n-1).

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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.

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A352932 Where the parity of A352931 changes.

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A354731 Absolute values of first differences of A354688.

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A372975 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) shares a factor with a(n-1) while omega(a(n)) does not equal omega(a(n-1)).

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A372974 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) is coprime to a(n-1) and omega(a(n)) does not equal omega(a(n-1)).

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Mathematica

A370500 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) does not share a factor with a(n-1) but sopfr(a(n)) does share a factor with soprf(a(n-1)), where sopfr(k) is the sum of the primes dividing k, with repetition.

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