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A356903 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous tau(a(n)) terms.

%I A356903 #14 Jan 16 2023 09:10:46
%S A356903 1,2,3,5,7,4,9,11,13,17,8,15,19,23,29,14,25,27,31,37,22,35,39,41,43,
%T A356903 34,47,21,53,55,26,49,51,59,61,10,67,33,71,73,38,65,69,77,79,58,83,57,
%U A356903 85,89,46,91,87,95,97,62,101,103,81,107,74,109,113,93,115,82,119,121,111,125,86,127,131,123
%N A356903 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous tau(a(n)) terms.
%C A356903 The terms are concentrated along various lines that contain numbers with a lowest prime factor of 2, 3 or 5. These lines appear to have a slight upward curvature. However the uppermost line, which has a gradient of ~1.22, contains numbers with all prime factors. See the linked images.
%C A356903 Numbers with a large number of divisors relative to the numbers close to it appear much later in the sequence. For example a(96) = 6, a(1873) = 12, a(2328) = 18, a(192) = 16. The sequence is conjectured to be a permutation of the positive integers although it may take a very large number of terms for some values to appear, e.g., after 500000 terms numbers such as 24, 30, 36 have not occurred. In the same range the longest run of consecutive odd values is seven, while the only fixed points are the first three terms, although it is possible others exist for very large values of n if the smaller terms continue to increase relative to the uppermost line.
%H A356903 Scott R. Shannon, <a href="/A356903/a356903_2.png">Image of the first 500000 terms</a>.
%H A356903 Scott R. Shannon, <a href="/A356903/a356903_3.png">Image of the first 500000 terms with color</a>. The terms with a lowest prime factor of 2 are shown in red, those with 3 are shown in yellow, those with 5 are shown in green, while all others are shown in white.
%e A356903 a(7) = 9 as tau(9) = A000005(9) = 3, and 9 is coprime to the previous three terms, namely a(6) = 4, a(5) = 7 and a(4) = 5.
%Y A356903 Cf. A356850, A000005, A356851, A093714, A336957, A000040.
%K A356903 nonn
%O A356903 1,2
%A A356903 _Scott R. Shannon_, Sep 03 2022