A375563 -id:A375563 - cp's OEIS frontend

A375564 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that is coprime to a(n-1) if a(n-1) is prime, otherwise a(n) shares a factor with a(n-1).

1, 2, 3, 4, 6, 8, 10, 5, 7, 9, 12, 14, 16, 18, 15, 20, 22, 11, 13, 17, 19, 21, 24, 26, 28, 30, 25, 35, 40, 32, 34, 36, 27, 33, 39, 42, 38, 44, 46, 23, 29, 31, 37, 41, 43, 45, 48, 50, 52, 54, 51, 57, 60, 55, 65, 70, 49, 56, 58, 62, 64, 66, 63, 69, 72, 68, 74, 76, 78, 75, 80, 82, 84, 77, 88, 86, 90

Offset: 1

Scott R. Shannon, Aug 19 2024

nonn

The sequence contains groups of consecutive primes separated by groups of composite terms. See the attached images of the first 100000 and 25 million  terms.
The fixed points are 1, 2, 3, 4, 15, 51, 63, 363, 437, ... There are thirty-two fixed points in the first 100000 terms and it is likely there are infinitely many in total. The sequence is conjectured to be a permutation of the positive integers.
Theorem: This sequence is a permutation of the positive integers, and the primes appear in their natural order. See link for proof. - N. J. A. Sloane, Sep 24 2024 and Oct 02 2024

			a(8) = 7 as a(7) = 5 is a prime number, and 7 is the smallest unused number that is coprime to 5.
Comment from _N. J. A. Sloane_, Oct 02 2024 (Start)
The following table was calculated by _Scott R. Shannon_ on Sep 29 2024. It shows the beginning, end, and length of the k-th run of successive primes.
  a  b c  : d e f [a = k, b = A376192(k), c = A376193(k),
  1  2 2  : 3 3 2   d = A376196, e = A376197, f = A376195]
  2  8 5  : 9 7 2
  3  18 11  : 21 19 4
  4  40 23  : 45 43 6
  5  84 47  : 92 83 9
  6  162 89  : 177 167 16
  7  321 173  : 351 349 31
  8  649 353  : 702 683 54
  9  1286 691  : 1379 1361 94
  10  2550 1367  : 2724 2699 175
  11  5096 2707  : 5412 5387 317
  12  10188 5393  : 10787 10739 600
  13  20406 10753  : 21502 21467 1097
  14  40883 21481  : 42958 42863 2076
  15  81932 42899  : 85791 85717 3860
  16  164190 85733  : 171441 171103 7252
  17  328490 171131  : 342216 341729 13727
  18  657509 341743  : 683462 682811 25954
  19  1316258 682819  : 1365580 1364747 49323
  20  2635513 1364761  : 2729447 2728129 93935
  21  5276876 2728163  : 5456194 5454167 179319
  22  10565366 5454181  : 10908253 10906271 342888
  23  21155215 10906297  : 21812343 21808453 657129
  24  42355195 21808483  : 43616683 43611721 1261489
  25  84797387 43611731  : 87223016 87215467 2425630
  26  169759097 87215483  : 174430392 174419101 4671296
(End)

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, with a color function showing primes in red, perfect powers of primes in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, with purple additionally representing powerful numbers that are not prime powers.
Scott R. Shannon, Image of the first 100000 terms. The terms are colored red, yellow, green, blue, violet if they have one, two, three, four, or five or more prime factors. The thin white line is a(n) = n.
Scott R. Shannon, Image of the first 25 million terms [Conventions as in preceding image.]
N. J. A. Sloane, Proof that this sequence is a permutation of the positive integers
N. J. A. Sloane, Table of n, a(n) for n = 1..100000
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A375563, A064413, A000040, A373546, A373545.

See also A376189, A376191-A376197, A376584.

nn = 120; c[_] := False; Do[Set[{a[i], c[i]}, {i, True}], {i, 2}];
j = a[2]; u = 3;
Do[k = u; If[PrimeQ[j],
    While[Or[c[k], GCD[j, k] > 1], k++],
    While[Or[c[k], CoprimeQ[j, k]], k++]];
  Set[{a[i], c[k], j}, {k, True, k}];
  If[k == u, While[c[u], u++]], {i, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Sep 29 2024 *)

from itertools import islice
from math import gcd
from sympy import isprime
def A375564_gen(): # generator of terms
    aset, a, b = {1,2}, 2, 3
    yield from (1,2)
    while True:
        c = b
        if isprime(a):
            while c in aset or gcd(c,a)>1:
                c+=1
        else:
            while c in aset or gcd(c,a)==1:
                c+=1
        aset.add(c)
        yield (a:=c)
        while b in aset:
            b += 1
A375564_list = list(islice(A375564_gen(),20)) # Chai Wah Wu, Sep 30 2024

A379248 a(1) = 1, a(2) = 2, for a(n) > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) while no exponent of each distinct prime factor of a(n) is the same as the exponent of the same prime factor of a(n-1).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 9, 3, 18, 15, 25, 5, 50, 16, 14, 20, 22, 24, 26, 28, 30, 27, 21, 36, 32, 34, 40, 38, 44, 42, 45, 33, 54, 39, 63, 48, 46, 52, 56, 49, 7, 98, 35, 75, 55, 100, 58, 60, 62, 64, 66, 68, 70, 72, 51, 81, 57, 90, 69, 99, 78, 76, 74, 80, 82, 84, 86, 88, 92, 94, 96, 104, 102, 108, 87, 117, 93, 126, 111, 135, 114, 112, 106, 116, 110, 121, 11, 242

Offset: 1

Scott R. Shannon, Dec 18 2024

For the terms studied the primes appear as terms in their natural order, and when a prime p appears as a term, the preceding term is always p^2 and the following term is always 2*p^2; it is likely this is true for all primes. A similar pattern is seen in the EKG sequence A064413 except that there a prime is always preceded by 2*p and followed by 3*p.
Unlike the EKG sequence a prime can appear as a factor of a preceding term long before it appears as a term by itself - see A379291 for the indices where each prime first appears as a factor of a(n).
The indices where the primes appear show an interesting pattern of runs of consecutive primes that are separated by only 6 terms, with longer, sometimes much longer, gaps in between - see A379290 and A379296. These primes appear in regions where the terms overall show a strong oscillating pattern of jumping between terms containing a prime p and p^2 as a factor. The primes being oscillated between increase until a new prime q appears in a term q^2 which leads to the next term being q. The occurrence of a new prime q can start a run of consecutive primes appearing before these oscillations subside and the terms slowly grow again until the next oscillation. See the attached graphs which show the burst/oscillating behavior, with the primes appearing in these regions, followed by terms with a slow, more linear, growth.
In the first 500000 terms there are only six fixed points - see A379292. However, as the regions of oscillating terms crosses the a(n) = n line it is likely more exist for larger values of n.
The sequence is conjectured to be a permutation of the positive integers. See A379293 for the index where n first appears.

			a(3) = 4 as 4 is unused and shares a factor with a(2) = 2, while 4 = 2^2 which has 2 as the exponent of the prime 2, while a(2) = 2^1 which has exponent 1. As these are different 4 is acceptable.
a(5) = 8 as 8 is unused and shares a factor with a(4) = 6, while 8 = 2^3 which has 3 as the exponent of the prime 2, while a(4) = 2^1*3^1 which has exponent 1. As these are different 8 is acceptable. Note that although 3 shares a factor with 6, 3 = 3^1 which has the same exponent 1 on the prime 3 as 6 = 2^1*3^1, so 3 cannot be chosen. This is the first term to differ from A064413.

Scott R. Shannon, Table of n, a(n) for n = 1..20000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..65536.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..16384, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither prime powers nor squarefree numbers in both blue and purple, where purple represents powerful numbers that are not prime powers.
Michael De Vlieger, Plot p^m | a(n) at (x, y) = (n, pi(p)), n = 1..2048, 4X vertical exaggeration, with a color function showing m = 1 in black, m = 2 in red, m = 3 in orange, ..., m = 11 in magenta.
Scott R. Shannon, Image of the first 500000 terms. The green line is a(n) = n.
Scott R. Shannon, Colored image of the first 10000 terms. The terms with one, two, three,... as their maximum prime factorization exponent are colored red, orange, yellow,... . The green line is a(n) = n.

Cf. A124010, A027746, A051903, A064413, A348086, A375564, A375563, A373546, A373545.

Cf. A379290 (index where prime n appears as a term), A379296 (differences between indices where prime terms appear), A379291 (index where prime n first appears as a factor of a(n)), A379293 (index where n appears as a term), A379292 (fixed points), A379294 (record high values), A379295 (indices of record high values).

nn = 120; c[_] := False;
f[x_] := f[x] = FactorInteger[x]; j = 2; u = 3;
{1, 2}~Join~Reap[Do[
  k = u; While[Or[c[k], CoprimeQ[j, k], AnyTrue[f[k], MemberQ[f[j], #] &]], k++];
    Set[{j, c[k]}, {k, True}]; Sow[k];
If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, 1]] (* Michael De Vlieger, Dec 21 2024 *)

from sympy import factorint
from itertools import islice
from collections import Counter
fcache = dict()
def myfactors(n):
    global fcache
    if n in fcache: return fcache[n]
    ans = Counter({p:e for p, e in factorint(n).items()})
    fcache[n] = ans
    return ans
def agen(): # generator of terms
    yield 1
    an, a, m = 2, {1, 2}, 3
    while True:
        yield an
        k, fan = m-1, myfactors(an)
        sfan = set(fan)
        while True:
            k += 1
            if k in a: continue
            fk = myfactors(k)
            sfk = set(fk)
            if sfk & sfan and all(fk[p]!=fan[p] for p in sfan):
                an = k
                break
        a.add(an)
print(list(islice(agen(), 89))) # Michael S. Branicky, Jan 05 2025

A377157 a(n) = n for n <= 3; for n > 3, a(n) is the smallest unused positive number that is coprime to a(n-1) if a(n-1) is prime and coprime to a(n-2) if a(n-2) is prime, else a(n) shares a factor with a(n-1) and a(n-2) if both are composite.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 10, 12, 14, 16, 18, 20, 15, 24, 9, 21, 27, 30, 33, 22, 11, 7, 13, 17, 19, 23, 25, 26, 40, 28, 32, 34, 36, 38, 42, 44, 46, 48, 50, 45, 35, 55, 60, 65, 39, 52, 54, 56, 58, 62, 64, 66, 68, 51, 72, 57, 63, 69, 75, 78, 70, 74, 76, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 85, 105, 95, 110, 114, 99, 81, 87, 93, 108, 111, 117, 120, 104

Offset: 1

Scott R. Shannon, Oct 18 2024

nonn

Similar to A375564 the primes appear in clusters separated by runs of composites. However unlike A375564 the runs of primes are interrupted by occasional composites, typically toward the end of the run - see the attaching image of the first 5000 terms. The first prime in the runs can also be larger than one or more subsequent primes, so unlike A375564, the primes do not all occur in their natural order; in the first 3000000 terms the out-of-order primes are 11 and 3581.
The runs of primes also appear to occur much less frequently than in A375564 - the last known run begins at a(156233) = 3581 and ends at a(169394) = 146857, and no more primes appear up to 3000000 terms.
The fixed points begin 1, 2, 3, 6, 32, 57, 119, 343, 3123, 3859, 120481. The sequence is conjectured to be a permutation of the positive integers.

			a(5) = 4 as both a(3) = 3 and a(4) = 5 are prime so a(5) must be coprime to both, and 4 is the smallest unused number coprime to both 3 and 5.
a(6) = 6 as a(4) = 5 is prime so a(6) must be coprime to a(4), and 6 is the smallest unused number coprime to 5.
a(7) = 8 as both a(5) = 4 and a(6) = 6 are composite so a(7) must share a factor with both, and 8 is the smallest unused number that shares a factor with both 4 and 6.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 5000 terms. The terms are colored red, yellow, green, blue, violet if they have one, two, three, four, or five or more prime factors. The thin white line is a(n) = n.
Scott R. Shannon, Image of the first 1000000 terms.

Cf. A375564, A375563, A098550, A336957, A000040.

A384045 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) if it is greater than it, else it is coprime to a(n-1) if it is less than it.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 7, 14, 9, 8, 12, 11, 22, 13, 26, 15, 18, 17, 16, 20, 19, 38, 21, 24, 23, 46, 25, 30, 29, 27, 33, 28, 32, 31, 62, 35, 34, 36, 39, 37, 74, 41, 40, 42, 44, 43, 86, 45, 48, 47, 94, 49, 56, 51, 50, 52, 54, 53, 106, 55, 60, 59, 57, 63, 58, 64, 61

Offset: 1

Scott R. Shannon, May 18 2025

nonn

For the terms studied all primes appear in their natural order, and approximately 65% of all primes p are immediately followed by a term 2*p. These later terms form the upper of the two lines in the graph.
In the first 100000 terms the fixed points are 1, 2, 12, 18, 98, 182, 306, 380; it is likely no more exist.
The sequence is a permutation of the positive integers as the lowest unused number after k terms will always appear as it will eventually be coprime to a(j) for some j > k.

			a(3) = 4 as a(2) = 2 and 4 > 2 and shares a factor with it. Note 3 cannot be chosen as 3 > 2 but is coprime to 2.
a(4) = 3 as a(3) = 4 and 3 < 4 and is coprime to it.

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

Cf. A375563, A375564, A373545, A373546, A064413.

nn = 120; c[_] := False; j = 2; u = 3; c[1] = c[2] = True;
{1, 2}~Join~Reap[Do[k = u;
  While[And[k < j, Or[c[k], ! CoprimeQ[j, k]]], k++];
    If[k >= j,
      If[PrimePowerQ[j],
        Set[{p, k}, {FactorInteger[j][[1, 1]], 1}]; While[c[k*p], k++]; k *= p,
        While[Or[c[k], CoprimeQ[j, k]], k++] ] ];
    Sow[k]; Set[{c[k], j}, {True, k}];
    If[k == u, While[c[u], u++]],
{n, 3, nn}] ][[-1, 1]] (* Michael De Vlieger, May 27 2025 *)

cp's OEIS Frontend

A375564 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that is coprime to a(n-1) if a(n-1) is prime, otherwise a(n) shares a factor with a(n-1).

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A379248 a(1) = 1, a(2) = 2, for a(n) > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) while no exponent of each distinct prime factor of a(n) is the same as the exponent of the same prime factor of a(n-1).

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A377157 a(n) = n for n <= 3; for n > 3, a(n) is the smallest unused positive number that is coprime to a(n-1) if a(n-1) is prime and coprime to a(n-2) if a(n-2) is prime, else a(n) shares a factor with a(n-1) and a(n-2) if both are composite.

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A384045 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) if it is greater than it, else it is coprime to a(n-1) if it is less than it.

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