A337687 -id:A337687 - cp's OEIS frontend

A353916 a(1) = 1, a(2) = 2. Let j = a(n-1) and let m = omega(gcd(j, k)) with gcd(j, k) > 1. For n > 2, a(n) = least k such that min(omega(j), omega(k)) = m and m < max(omega(j), omega(k)).

1, 2, 6, 3, 12, 4, 10, 5, 15, 9, 18, 8, 14, 7, 21, 27, 24, 16, 20, 25, 30, 32, 22, 11, 33, 66, 36, 42, 28, 49, 35, 70, 40, 60, 45, 81, 39, 13, 26, 64, 34, 17, 51, 102, 48, 78, 52, 128, 38, 19, 57, 114, 54, 84, 56, 126, 63, 105, 75, 90, 50, 110, 44, 121, 55, 125

Offset: 1

Michael De Vlieger, May 10 2022

nonn

Let P = the set of distinct prime divisors of j = a(n-1), and let Q = the set of distinct prime divisors of k = a(n). Let g = gcd(j, k) > 1 and let G = {P intersect Q}. Noncoprime j and k implies |G| > 0. This sequence is such that |G| > 0, |P| > |G|, and |Q| == |G|, or vice versa.
Coprimality and equality are forbidden, forcing primes into divisibility. Because of this, a(i) = mp -> a(i+1) = p -> a(i+2) = sp, with composite m and s not powers of p > 2. For n > i, multiples of p including powers may appear in the sequence. Consequently, odd primes enter the sequence late.
Numbers m that immediately precede and follow prime p have omega(m) > 1.
For any adjacent pair of terms with n > 1, if one is prime then the other cannot be a power of that prime, since such a pair would have the same number of distinct prime divisors.
This sequence requires an asymmetric version of the relation of j and k seen in A337687. In that sequence, we have P != Q, |P| > |G| and |Q| > |G|, therefore we have symmetry in that there is at least 1 prime p | j that does not divide k, and at least 1 prime q | k that does not divide j. That sequence occurs among composites m with omega(m) > 1, but this sequence admits primes, since, for |G| = 1, we must have |P| or |Q| equal to 1, and there is no prohibition for multiplicity to exceed 1.
A353917 is a version of this sequence that prohibits divisibility, hence primes do not appear, but composite prime powers do.
Open question: do the primes appear in order? (They do for n <= 2^16).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^12, showing records in red and local minima in blue, highlighting fixed points in gold and primes in green.
Michael De Vlieger, Prime power factor study of a(n), n = 1..2^13 where n increases from left to right, and pi(p) from bottom to top in the top image. Multiplicity e of p^e is represented by black for e = 1, red for e = 2, and a color function to the maximum multiplicity in the range thereafter. On the lower bar we show primes in red, composite prime powers in yellow, squarefree numbers m with omega(m) > 1 in green, and nonsquarefree composite numbers m with omega(m) > 1 (i.e., in A126706) in blue.
Index entries for sequences related to EKG sequence

Cf. A064413, A337687, A353917.

nn = 2^8; c[_] = 0; a[1] = c[1] = 1; j = a[2] = c[2] = 2; u = 3; Do[Set[k, u]; Set[m, PrimeNu[j]]; While[Nand[c[k] == 0, And[#2 > #3, #1 == #3] & @@ Append[Sort[{m, PrimeNu[k]}], PrimeNu[GCD[j, k]]]], k++]; Set[{a[i], c[k]}, {k, i}]; j = k; If[k == u, While[c[u] > 0, u++]], {i, 3, nn}]; Array[a, nn]

A353917 a(1) = 4. Let j = a(n-1) and let m = omega(gcd(j, k)) with gcd(j, k) > 1. For n > 1, a(n) = least k such that min(omega(j), omega(k)) = m and m < max(omega(j), omega(k)), but neither j | k nor k | j.

Original entry on oeis.org

4, 6, 8, 10, 16, 12, 9, 15, 25, 20, 30, 18, 27, 21, 49, 14, 32, 22, 64, 24, 42, 28, 70, 40, 60, 36, 66, 44, 110, 50, 90, 48, 78, 52, 128, 26, 169, 39, 81, 33, 121, 55, 125, 35, 343, 56, 84, 54, 102, 68, 170, 80, 120, 45, 105, 63, 168, 72, 114, 76, 190, 100, 130

Offset: 1

Michael De Vlieger, May 10 2022

nonn

The sequence exhibits phases involving alternating composite prime powers and squarefree semiprimes. These manifest in log-log scatterplot in a caustic fashion, where the composite prime power is very much larger than the squarefree semiprime for sufficiently large n.
Let P = the set of distinct prime divisors of j = a(n-1), and let Q = the set of distinct prime divisors of k = a(n). Let g = gcd(j, k) > 1 and let G = {P intersect Q}. Noncoprime j and k implies |G| > 0. This sequence is such that |G| > 0, |P| > |G|, and |Q| == |G|, or vice versa, yet neither j | k nor k | j.
Theorem: primes are prohibited. Proof: since we have gcd(j, k) > 1 and do not allow divisibility, and since primes must either divide or be coprime to another number m, primes do not appear in this sequence.
Theorem: squarefree semiprimes j = pq are followed by k = p^2 or k = q^2. Proof: since omega(j) = |P| = 2 and is squarefree, we have 2 cases pertaining to successor k, both with gcd(j, k) > 1.
1.) |P| == |G| implies |Q| > |G| and |Q| > |P|.
2.) |Q| == |G| implies |P| > |G| and |P| > |Q|.
The first case implies some prime r | k yet gcd(j, r) = 1. But this would require j | k, which is prohibited. The second case suggests either p | k or q | k, but so as to satisfy non-divisibility axiom, we are forced into either k = p^e, e > 1, or k = q^m, m > 1.
Corollary: powers of the same prime appear in natural order in this sequence.
There is a weaker alternation between numbers in A120944 and A350352 as n is sufficiently large. This alternation exhibits prime power factor features akin to the composite prime power-squarefree semiprime alternation.
Conjecture: permutation of composite numbers.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..1192, showing records in red and local minima in blue.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16, showing records in red, local minima in blue, accentuating composite prime powers in green and squarefree semiprimes in gold.
Michael De Vlieger, Prime power factor study of a(n), n = 1..10^4 where n increases from left to right, and pi(p) from bottom to top in the top image. Multiplicity e of p^e is represented by black for e = 1, red for e = 2, and a color function to the maximum multiplicity in the range thereafter. On the lower bar we show composite prime powers in yellow, squarefree semiprimes in orange, numbers in A350352 in green, and those in A126706 in blue.
Index entries for sequences related to EKG sequence

Cf. A006881, A064413, A126706, A246547, A337687, A353916.

nn = 2^7; c[_] = 0; j = a[1] = 4; c[4] = 1; u = 6; Do[Set[k, u]; Set[m, PrimeNu[j]]; While[Nand[c[k] == 0, ! Divisible[#2, #1] & @@ Sort[{j, k}], And[#2 > #3, #1 == #3] & @@ Append[Sort[{m, PrimeNu[k]}], PrimeNu[GCD[j, k]]]], k++]; Set[{a[i], c[k]}, {k, i}]; j = k; If[k == u, While[Nand[c[u] == 0, CompositeQ@ u], u++]], {i, 2, nn}]; Array[a, nn]

A362754 a(1) = 1, a(2) = 6; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and also contains as a factor the smallest prime that is not a factor of a(n-1).

Original entry on oeis.org

1, 6, 10, 12, 15, 18, 20, 24, 30, 14, 21, 28, 36, 40, 42, 35, 50, 45, 48, 60, 56, 54, 70, 63, 66, 55, 22, 33, 44, 72, 75, 78, 65, 26, 39, 52, 84, 80, 90, 98, 96, 100, 102, 85, 34, 51, 68, 108, 105, 110, 99, 88, 114, 95, 38, 57, 76, 120, 112, 126, 130, 117, 104, 132, 135, 138, 115, 46, 69, 92, 144

Offset: 1

Scott R. Shannon, May 02 2023

nonn

No term can be a prime power as each term must contain at least two distinct prime factors. This make the sequence similar to A360519 and A361606. A close examination of the lines of concentrated terms, see the attached images, shows they have a slight downward curvature. In the first 250000 terms the only fixed points are 1, 69, 87, 116825, although it is possible more exist for very large values of n.

			a(3) = 10 as a(2) = 6 = 2*3, and 10 is the smallest unused number that shares a factor with 6 while also containing 5 as a prime factor, the smallest prime not a factor of 6.
a(4) = 12 as a(3) = 10 = 2*5, and 12 is the smallest unused number that shares a factor with 10 while also containing 3 as a prime factor, the smallest prime not a factor of 10.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing squarefree composites in green, numbers neither squarefree nor prime powers in blue, and numbers in A286708 in large light blue.
Scott R. Shannon, Image of the first 250000 terms. The green line is a(n) = n.
Scott R. Shannon, Image of the first 250000 terms in color. Terms with a lowest prime factor 2, 3, 5, 7, 11, >=13 are colored white, red, yellow, green, blue, violet and light gray respectively.

Cf. A337687, A351495, A064413, A360519, A361606.

nn = 120; c[] := False; m[] := 2; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, {1, 6}]; j = a[2]; Do[q = 2; While[Divisible[j, q], q = NextPrime[q]]; k = m[q]; While[Or[c[#], PrimePowerQ[#], CoprimeQ[j, k]] &[q k], k++]; k *= q; While[c[m[q] q], m[q]++]; Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, May 02 2023 *)

A362600 a(1) = 1, a(2) = 6, a(3) = 10; for n > 3, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and a(n-2) and also contains as factors the smallest primes that are not factors of both a(n-1) and a(n-2).

Original entry on oeis.org

1, 6, 10, 15, 12, 20, 30, 42, 35, 40, 60, 84, 70, 45, 18, 50, 75, 24, 80, 90, 105, 14, 36, 120, 140, 21, 48, 150, 210, 154, 33, 54, 110, 135, 66, 100, 165, 72, 130, 180, 126, 175, 160, 168, 195, 170, 78, 225, 190, 96, 240, 280, 63, 102, 270, 315, 28, 108, 300, 350, 147, 114, 330, 420, 77, 22

Offset: 1

Scott R. Shannon, May 02 2023

nonn

No term can be a prime power as each term must contain at least two distinct prime factors. This make the sequence similar to A362754, A360519 and A361606. Some small composite numbers take many terms to appear, e.g., a(354476) = 65. Such terms are usually preceded by a term that contains all the lower primes as factors. In the first 500000 terms, other than the first term, there are no fixed points, and it is unknown if any exist.

			a(4) = 15 as a(2) = 6 = 2*3 and a(3) = 10 = 2*5, and 15 is the smallest unused number that shares a factor with 6 and 10 while also containing 5 and 3 as prime factors, the smallest primes not factors of 6 and 10 respectively. This is the first term to differ from A362754.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing squarefree numbers in green and nonsquarefree numbers in blue, highlighting nonsquarefree numbers that are powerful (in A001694) with large light blue dots.
Scott R. Shannon, Image of the first 250000 terms. The green line is a(n) = n.
Scott R. Shannon, Image of the first 250000 terms in color. Terms with a lowest prime factor 2, 3, 5, 7, 11, >=13 are colored white, red, yellow, green, blue, violet and light gray respectively.

Cf. A362754, A337687, A351495, A064413, A360519, A361606, A053669.

nn = 120; c[_] := False;
f[x_] := If[OddQ[x], 2, y = 3; While[Divisible[x, y], y = NextPrime[y]]; y];
MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {1, 6, 10}];
i = a[2]; j = a[3]; q = 5; u = 12;
Do[qq = f[j]; k = Ceiling[u/#] &[q*qq];
  While[Or[c[#], CoprimeQ[i, #], CoprimeQ[i, j]] &[k*q*qq], k++];
  k *= q*qq;
  Set[{a[n], c[k], i, j, q}, {k, True, j, k, qq}];
  If[k == u, While[Or[c[u], PrimePowerQ[u]], u++]], {n, 4, nn}];
Array[a, nn] (* Michael De Vlieger, May 09 2023 *)

A339316 a(1) = 2; for n > 1, a(n) = smallest composite number not occurring earlier which does not share a factor with a(n-1).

Original entry on oeis.org

2, 9, 4, 15, 8, 21, 10, 27, 14, 25, 6, 35, 12, 49, 16, 33, 20, 39, 22, 45, 26, 51, 28, 55, 18, 65, 24, 77, 30, 91, 32, 57, 34, 63, 38, 69, 40, 81, 44, 75, 46, 85, 36, 95, 42, 115, 48, 119, 50, 87, 52, 93, 56, 99, 58, 105, 62, 111, 64, 117, 68, 121, 54, 125, 66, 133, 60, 143, 70, 123, 74, 129

Offset: 1

Scott R. Shannon, Nov 30 2020

nonn

The sequence excludes primes as otherwise the terms would simply be all the ordered integers >= 2. The terms appear to cluster around two lines; the lower line is a(n) ~ n while the upper lines starts with a gradient of approximately 2 and then slowly flattens. It is possible this gradient approaches 1 as n->infinity.

			a(2) = 9, as a(1) = 2 thus a(2) cannot contain 2 as a factor and cannot be a prime. The lowest unused composite matching these criteria is 9.
a(3) = 4, as a(2) = 9 and thus a(3) cannot contain 3 as a factor and cannot be a prime. The lowest unused composite matching these criteria is 4.
a(4) = 15, as a(3) = 4 and thus a(4) cannot contain 2 as a factor and cannot be a prime. The lowest unused composite matching these criteria is 15.

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

Cf. A337687, A000961, A064413, A336957, A098550, A020639, A280864.

isok(k, fprec, v) = {if (!isprime(k) && #select(x->(x==k), v) == 0, #setintersect(Set(factor(k)[,1]), fprec) == 0;);}
lista(nn) = {my(va= vector(nn)); va[1] = 2; for (n=2, nn, my(k=2, fprec = Set(factor(va[n-1])[,1])); while (! isok(k, fprec, va), k++); va[n] = k;); va;} \\ Michel Marcus, Nov 30 2020

from sympy import isprime, primefactors as pf
def aupton(terms):
  alst, aset = [2], {2}
  for n in range(2, terms+1):
    m, prevpf = 4, set(pf(alst[-1]))
    while m in aset or isprime(m) or set(pf(m)) & prevpf != set(): m += 1
    alst.append(m); aset.add(m)
  return alst
print(aupton(72)) # Michael S. Branicky, Feb 09 2021

A352098 a(1) = 6, a(2) = 15; let i = a(n-2) and j = a(n-1); a(n) = least k not already in the sequence such that gcd(j, k) = 1 and gcd(i, k) = m > 1 and both omega(i) and omega(k) exceed omega(m), where omega = A001221.

Original entry on oeis.org

6, 15, 14, 33, 10, 21, 22, 35, 12, 55, 26, 45, 28, 39, 20, 51, 38, 63, 34, 57, 40, 69, 44, 75, 46, 65, 18, 85, 52, 95, 24, 115, 56, 135, 58, 93, 50, 87, 62, 99, 68, 77, 30, 91, 66, 119, 60, 133, 74, 105, 76, 111, 70, 117, 82, 129, 80, 123, 86, 141, 88, 147, 92

Offset: 1

Michael De Vlieger, Jun 03 2022

nonn

Variant of A098550 analogous to A337687 regarding its relationship to A064413.
Theorem: the sequence contains numbers in A024619. Proof: with m = gcd(i, k) > 1, we have omega(m) >= 1, however, both omega(i) and omega(k) must exceed omega(m). Therefore, if no number in the sequence has a single distinct prime factor, none can arise. This restricts the sequence to numbers that are not prime powers.
Theorem: i and k are nondivisors of one another. Proof: both omega(i) and omega(k) exceed omega(m), therefore there exists some prime p | i and some prime q | k, yet, p does not divide k and q does not divide i. Hence i does not divide k and k does not divide i.
The numbers i and k have the relationship described in A272619, a sort of relationship described in A045763.
Theorem: if prime p | j then p does not divide k. Consequence of coprimality axiom gcd(j, k) = 1. Hence, even terms are nonadjacent in the sequence. Therefore we begin this sequence with {6, 15}.
Composite quasi-rays in the Yellowstone sequence scatterplot are retained, bifurcated according to parity for same reasons as in that sequence.
Conjecture: permutation of A024619.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^10, showing records in red and local minima in blue, highlighting even terms in gold.
Michael De Vlieger, Prime power factor diagram for p^e | a(n), n = 1..320, where the upper portion plots p^e at (n, pi(p)), with a color function representing e where black = 1, red = 2, etc. to magenta representing the largest e in the range. The lower portion uses green to represent squarefree a(n) and blue to represent other composite a(n).

Cf. A001221, A024619, A045763, A064413, A098550, A337687.

c[_] = 0; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {6, 15}]; Set[{i, j, u, nn}, {a[1], a[2], 10, 120}]; Do[k = u; m = PrimeNu[i]; While[Nand[c[k] == 0, And[# > 1, And[m > #, PrimeNu[k] > #] &@ PrimeNu[#]] &@ GCD[i, k], CoprimeQ[j, k]], k++]; Set[{a[n], c[k], i, j}, {k, n, j, k}]; If[k == u, While[Nand[c[u] == 0, ! PrimePowerQ[u]], u++]], {n, 3, nn}]; Array[a, nn]

A362842 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) when both a(n-1) and a(n) are read as numbers in bases from one more than the maximum digit in a(n-1) and a(n), up to base 10.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 12, 24, 8, 20, 10, 30, 33, 11, 22, 26, 13, 39, 15, 48, 28, 14, 49, 7, 70, 16, 38, 19, 57, 69, 18, 56, 76, 36, 60, 40, 42, 21, 63, 66, 44, 46, 23, 92, 32, 64, 62, 31, 93, 27, 90, 5, 50, 55, 77, 84, 35, 80, 68, 17, 119, 34, 94, 47, 329, 91, 52, 96, 45, 95, 25, 190, 54, 98, 58, 29

Offset: 1

Scott R. Shannon, May 05 2023

This is a base variation of the EKG sequence A064413. Despite numbers with larger digits having to share a factor with a(n-1) in fewer bases than those with only small digits, and would therefore seemingly appear more frequently, the frequency of the digits 8 and 9, for example, in the first 200000 terms is the same as the smaller digits 0 to 7, so surprisingly this does not appear to influence the determination of a(n).

In the first 200000 terms the smallest unused number is 25411, which implies all numbers will eventually appear. In the same range the fixed points are 1, 2, 424, 507, 1261, 1577, 2461, 4311; it is likely no more appear.

			a(7) = 12 as the maximum digit in a(6) = 9 and 12 is 9, so a(6) and a(7) are only read as base 10 numbers, and 12 is the smallest unused number which shares a factor with 9 in base 10.
a(8) = 24 as the maximum digit in a(7) = 12 and 24 is 4, and 12_k shares a factor with 24_k when they are read as numbers in all bases k = 5,6,7,8,9,10. No unused smaller number has this property, e.g. a(8) cannot equal 8 as a(7) in base 9 is 12_9 = 11, which does not share a factor with 8_9 = 8. This is the first term to differ from A064413.
a(9) = 8 as the maximum digit in a(8) = 24 and 8 is 8, and 24_k shares a factor with 8_k when they are read as numbers in all bases k = 9,10.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Cf. A064413, A004053, A354087, A352763, A348086, A337687.

A381139 a(1) = 1, a(2) = 2. Let j = a(n-1) and let d = A160995(j) be the smallest non-divisor of j which shares a prime factor with j. Then for n > 2 a(n) is the smallest multiple of d which is not yet in the sequence.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 9, 30, 28, 36, 32, 42, 40, 48, 27, 54, 44, 60, 56, 66, 52, 72, 10, 64, 78, 68, 84, 80, 90, 76, 96, 45, 102, 88, 108, 104, 114, 92, 120, 63, 126, 100, 132, 112, 138, 116, 144, 50, 124, 150, 128, 156, 136, 162, 140, 168, 81, 174

Offset: 1

David James Sycamore and Michael De Vlieger, Feb 15 2025

2 is the only prime in the sequence. Initially every odd term > 1 is a multiple of 9.
The first 3 odd terms not divisible by 9 are the following: a(19241) = 25, a(38481) = 75, a(57719) = 125, a(76959) = 175, a(115438) = 275. Differences between indices of odd terms divisible by 25 but not 9 are approximately 19240.
In this sequence, 25 follows 55440 = 2^4 * 3^2 * 5 * 7 * 11. The number 49 is missing for n <= 2^24.
Though a(n+1) must neither divide a(n) nor be coprime to same, a(n) may divide a(n+1). Examples: the sequence begins with {1, 2, 4}, a(19) = 27 and a(20) = 54, a(44) = 63 and a(45) = 126, etc.

			a(3) = 4, the smallest non divisor of 2 which shares a divisor with 2.
a(4) = 6, the smallest non divisor of 4 which shares a divisor with 4.
a(5) = 8 because 4 is the smallest non divisor of 6 with a factor in common but 4 has been seen earlier and 8 is the least novel multiple of 4.
a(10) = 24 implies a(11) = 9 since A160995(24) = 9, and 9 has not occurred earlier. This is the first odd number in the sequence. Thereafter we see a(19) = 27, a(36) = 45, a(44) = 63, a(61) = 81, etc (consecutive odd multiples of 9). It is not known if this behavior continues as the sequence extends.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Prime power decomposition of a(n), n = 1..1000, (2025).

Cf. A160995, A337687.

nn = 120; c[] := False; m[] := 1; j = 2;
{1, 2}~Join~
  Reap[Do[k = 2; Until[Nor[Divisible[j, k], CoprimeQ[j, k]], k++];
    While[Or[c[#], Divisible[j, #], CoprimeQ[j, #]] &[k*m[k]], m[k]++];
      k *= m[k]; c[k] = True; j = Sow[k],
{n, nn}] ][[-1, 1]] (* Michael De Vlieger, Feb 18 2025 *)

cp's OEIS Frontend

A353916 a(1) = 1, a(2) = 2. Let j = a(n-1) and let m = omega(gcd(j, k)) with gcd(j, k) > 1. For n > 2, a(n) = least k such that min(omega(j), omega(k)) = m and m < max(omega(j), omega(k)).

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A353917 a(1) = 4. Let j = a(n-1) and let m = omega(gcd(j, k)) with gcd(j, k) > 1. For n > 1, a(n) = least k such that min(omega(j), omega(k)) = m and m < max(omega(j), omega(k)), but neither j | k nor k | j.

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A362754 a(1) = 1, a(2) = 6; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and also contains as a factor the smallest prime that is not a factor of a(n-1).

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A362600 a(1) = 1, a(2) = 6, a(3) = 10; for n > 3, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and a(n-2) and also contains as factors the smallest primes that are not factors of both a(n-1) and a(n-2).

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A339316 a(1) = 2; for n > 1, a(n) = smallest composite number not occurring earlier which does not share a factor with a(n-1).

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A352098 a(1) = 6, a(2) = 15; let i = a(n-2) and j = a(n-1); a(n) = least k not already in the sequence such that gcd(j, k) = 1 and gcd(i, k) = m > 1 and both omega(i) and omega(k) exceed omega(m), where omega = A001221.

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A362842 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) when both a(n-1) and a(n) are read as numbers in bases from one more than the maximum digit in a(n-1) and a(n), up to base 10.

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A381139 a(1) = 1, a(2) = 2. Let j = a(n-1) and let d = A160995(j) be the smallest non-divisor of j which shares a prime factor with j. Then for n > 2 a(n) is the smallest multiple of d which is not yet in the sequence.

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Mathematica