cp's OEIS Frontend

A356850 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 Omega(a(n)) terms.

%I A356850 #31 May 07 2025 13:16:04
%S A356850 1,2,3,5,4,7,9,10,11,13,6,17,19,14,15,23,22,21,25,26,29,27,31,8,33,35,
%T A356850 34,37,39,38,41,43,45,28,47,51,46,49,53,55,12,59,61,57,20,67,69,58,65,
%U A356850 71,62,63,73,74,77,75,79,52,83,85,81,44,89,87,82,91,93,86,95,97,94,99,101,103,50,107
%N A356850 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 Omega(a(n)) terms.
%C A356850 The terms are concentrated along lines that contain numbers with a lowest prime factor of 2 or 3. Two of these lines are initially separated but join after approximately 130000 terms. This combined line then joins the uppermost line, which contains numbers with all prime factors and has a gradient of ~1.59, after approximately 680000 terms at which point a new series of smaller values appears. See the linked images.
%C A356850 Numbers with a larger number of prime divisors relative to the numbers close to it appear much later in the sequence. For example a(4014) = 16, a(14219) = 40, while even more delayed are a(685301) = 24 and a(704634) = 36. These last two appear after the above mentioned line merging after 680000 terms.
%C A356850 The lower lines containing terms with prime factors of 2 and 3 visible in the image of terms up to 1000000 are curving upward, possibly repeating the earlier behavior seen where similar lines eventually join with the uppermost line. If these do in fact eventually reach the uppermost line it is plausible this will once again signal the start of a new series of much lower valued terms.
%C A356850 The two lowest unseen numbers after 1000000 terms are 32 and 48, for former indicating that the longest run of consecutive odd values is only four after 1000000 terms. Although the sequence is conjectured to be a permutation of the positive integers, if these missing terms, especially those of the form 2^k, only appear after the recombination of the lower lines with the upper line then it may take an extraordinarily large number of terms for some values, like 2^k for large k, to eventually appear.
%C A356850 In the first 1000000 terms the only fixed points are the first three terms along with 14 and 15. It is possible more exist if the above mentioned upward trend of smaller values does occur.
%H A356850 Michael De Vlieger, <a href="/A356850/b356850.txt">Table of n, a(n) for n = 1..16384</a>
%H A356850 Michael De Vlieger, <a href="/A356850/a356850_4.png">Annotated log-log scatterplot of a(n)</a> n = 1..2^15, showing records in red, local minima in blue, highlighting primes in green, other prime powers in gold, and numbers neither prime powers nor squarefree in light blue.
%H A356850 Michael De Vlieger, <a href="/A356850/a356850_5.png">Annotated log-log scatterplot of a(n)</a> n = 1..2^15, highlighting composite prime powers. Aside from composite 2^e, p^e appears early.
%H A356850 Michael De Vlieger, <a href="/A356850/a356850_6.png">Log log scatterplot of a(n)</a>, n = 1..2^15, with a color function according to Omega(a(n)), where red = 1, amber = 2, ..., dark blue = 6.
%H A356850 Scott R. Shannon, <a href="/A356850/a356850.png">Image for n=1..100000</a>. The green line is y = n.
%H A356850 Scott R. Shannon, <a href="/A356850/a356850_3.png">Image for n=1..100000 with color</a>. Terms with lowest prime factor of 2 are show in red, those with 3 in yellow, those with 5 in green, and all others in white.
%H A356850 Scott R. Shannon, <a href="/A356850/a356850_1.png">Image for n=1..1000000</a>.
%H A356850 Scott R. Shannon, <a href="/A356850/a356850_2.png">Image for n=1..1000000 with color</a>. The terms 24 and 36 can be seen on the x-axis at the bottom right in red.
%e A356850 a(7) = 9 as Omega(9) = A001222(9) = 2, and 9 is coprime to the previous two terms, namely a(6) = 7 and a(5) = 4.
%t A356850 nn = 120; c[_] = False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = 3; r = 2; m = 1; Do[k = u; If[# > m, m = #] &@ Floor@ Log2[r]; Do[If[i == 1, p[1] = a[n - i], Set[p[i], p[i - 1]*a[n - i]]], {i, m}]; While[Nand[! c[k], CoprimeQ[k, p[PrimeOmega[k]]]], k++]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]]; If[k > r, r = k], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger_, Nov 07 2022 *)
%Y A356850 Cf. A356903, A001222, A356851, A093714, A336957, A000040.
%K A356850 nonn
%O A356850 1,2
%A A356850 _Scott R. Shannon_, Aug 31 2022