A372974 -id:A372974 - cp's OEIS frontend

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)).

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 *)

A373357 a(1) = 1, a(2) = 2, a(3) = 15; for n > 3, a(n) is the smallest unused positive number that is coprime to a(n-1), shares a factor with a(n-2), while omega(a(n)) does not equal omega(a(n-1)) or omega(a(n-2)).

Original entry on oeis.org

1, 2, 15, 154, 3, 10, 231, 4, 21, 110, 7, 6, 385, 8, 33, 70, 9, 14, 165, 16, 35, 66, 5, 12, 455, 27, 20, 273, 25, 18, 595, 32, 45, 182, 81, 22, 105, 11, 24, 715, 64, 39, 140, 13, 28, 195, 49, 26, 315, 128, 51, 130, 17, 36, 935, 243, 34, 285, 256, 55, 42, 121, 38, 429, 19, 44, 399, 512, 57, 170

Offset: 1

Scott R. Shannon, Jun 02 2024

nonn

The sequence uses the same rules for selecting the next term as the Yellowstone permutation A098550 but with the additional restriction that the number of distinct prime factors of a(n) must be different to both a(n-1) and a(n-2). The terms show complicated behavior, being concentrated along various curved and straight lines some of which cross and some of which only have points for various ranges of n. See the attached images.

The fixed points begin 1, 2, 32, 51, although it is possible more exist. The sequence is likely to be a permutation of the positive integers.

			a(10) = 110 as 110 shares a factor with a(8) = 4, does not share a factor with a(9) = 21, while omega(110) = 3 does not equal omega(4) = 1 or omega(21) = 2.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, perfect powers of primes in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple, where purple additionally represents powerful numbers that are not prime powers.
Scott R. Shannon, Image of the first 5000 points. Numbers with one, two, three, or four and more distinct prime factors are show as red, yellow, green and violet respectively. The white line is a(n) = n.
Scott R. Shannon, Image of the first 200000 points.

Cf. A098550, A001221, A373350, A372975, A372974.

nn = 63; c[_] := False;
MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, {1, 2, 15}];
i = a[2]; j = a[3]; u = 3; v = 1; w = 2;
Do[k = u;
 While[Or[c[k],
   ! CoprimeQ[j, k],
   ! DuplicateFreeQ[{v, w, Set[x, PrimeNu[k]]}]],
 k++];
 Set[{a[n], c[k], i, j, v, w}, {k, True, j, k, w, x}];
 If[k == u, While[c[u], u++]], {n, 4, nn}];
Array[a, nn] (* Michael De Vlieger, Jun 09 2024 *)

cp's OEIS Frontend

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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