A375564 -id:A375564 - cp's OEIS frontend

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).

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

A375563 a(1) = 1; for n > 1, a(n) is the smallest unused positive number that shares a factor with a(n-1) if a(n-1) is prime, otherwise a(n) is coprime to a(n-1).

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 7, 14, 9, 8, 11, 22, 13, 26, 15, 16, 17, 34, 19, 38, 21, 20, 23, 46, 25, 12, 29, 58, 27, 28, 31, 62, 33, 32, 35, 18, 37, 74, 39, 40, 41, 82, 43, 86, 45, 44, 47, 94, 49, 24, 53, 106, 51, 50, 57, 52, 55, 36, 59, 118, 61, 122, 63, 64, 65, 42, 67, 134, 69, 56, 71, 142, 73, 146

Offset: 1

Scott R. Shannon, Aug 19 2024

nonn

The terms are all concentrated along three lines with the central line, consisting entirely of all the odd terms, along the line a(n) = n. Surprisingly however there are no fixed points in the first 100000 terms. In the same range the primes appear in their natural order. The sequence is conjectured to be a permutation of the positive integers.

			a(6) = 5 as a(5) = 6 is a composite number, and 5 is the smallest unused number that is coprime to 6.

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

Cf. A375564, A064413, A000040, A373546, A373545.

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 *)

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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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A375563 a(1) = 1; for n > 1, a(n) is the smallest unused positive number that shares a factor with a(n-1) if a(n-1) is prime, otherwise a(n) is coprime to 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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Mathematica