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)).

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

Author

Scott R. Shannon, Jun 02 2024

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)