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%I #11 Jun 02 2024 10:17:26
%S 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,
%T 22,102,16,24,105,17,26,110,19,28,114,23,33,120,25,34,126,27,35,130,
%U 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
%N 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)).
%C 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.
%C 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.
%C 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.
%H Scott R. Shannon, <a href="/A373350/b373350.txt">Table of n, a(n) for n = 1..10000</a>
%H Scott R. Shannon, <a href="/A373350/a373350.png">Image of the first 500 terms</a>. 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.
%H Scott R. Shannon, <a href="/A373350/a373350_1.png">Image of the first 50000 terms</a>.
%H Scott R. Shannon, <a href="/A373350/a373350_2.png">Image of the first 400000 terms</a>.
%e 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.
%Y Cf. A001221, A027748, A109465, A372975.
%K nonn
%O 1,2
%A _Scott R. Shannon_, Jun 01 2024