A373350 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that omega(a(n)) does not equal omega(a(n-1)) or omega(a(n-2)).

1, 2, 6, 30, 3, 10, 42, 4, 12, 60, 5, 14, 66, 7, 15, 70, 8, 18, 78, 9, 20, 84, 11, 21, 90, 13, 22, 102, 16, 24, 105, 17, 26, 110, 19, 28, 114, 23, 33, 120, 25, 34, 126, 27, 35, 130, 29, 36, 132, 31, 38, 138, 32, 39, 140, 37, 40, 150, 41, 44, 154, 43, 45, 156, 47, 46, 165, 49, 48, 168, 53, 50, 170, 59

Offset: 1

Author

Scott R. Shannon, Jun 01 2024

Keywords

• nonn

Comments

The terms of the sequence are initially concentrated along three lines, each line containing terms with either one, two, or three-or-more distinct prime factors. See the attached images. The two lower lines, containing terms with one and two distinct prime factors, cross near n = 66, and then the upper two, containing terms with one and three-or-more distinct prime factors, cross near n = 326. At this point the upper line changes and contains terms with one distinct prime factor but also terms with more than three distinct prime factors. After a large gap the lower two lines, containing terms with two and three distinct prime factors, cross near n = 25308, and then finally, after an even longer gap, the upper two lines, containing all terms except those with three distinct prime factors, merge and become one line near n = 344310. It is likely all subsequent terms fall into one of the two remaining lines, although this is unknown.

A number with one distinct prime factor appears as a term for all n with n mod 3 = 2 up until n = 350, when a(350) = 510 = 2*3*5*17 is smaller than the next unused number with one distinct prime factor. Likewise a number with three distinct prime factors appears as a term for all n with n mod 3 = 1 up until n = 100 when a(100) = 210 = 2*3*5*7 is smaller than the next unused number with three distinct prime factors. However, rather surprisingly, a number with two distinct prime factors appears as a term for all n with n mod 3 = 0 all the way up until n = 344307, when a(344307) = 373215 = 3*5*139*179 is smaller than the next unused number with two distinct prime factors. This value of n corresponds to the start of the merging of the upper two lines near n = 344310 described above.

The fixed points begin 1, 2, 15, 18, 125, 137, 2737, 120051; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

Examples

			a(4) = 30 as a(2) = 2 has one distinct prime factor and a(3) = 6 has two distinct prime factors, and 30 is the smallest unused number with three distinct prime factors.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 500 terms. In this and other images terms with one, two, three, or more than three distinct prime factors are shown in red, yellow, green, violet respectively. The white line is a(n) = n.
• Scott R. Shannon, Image of the first 50000 terms.
• Scott R. Shannon, Image of the first 400000 terms.

Crossrefs

Cf. A001221, A027748, A109465, A372975.