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

Author

Scott R. Shannon, May 26 2024

Keywords

• nonn

Comments

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)

Examples

			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.
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)