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A373698 a(n) = n for n <= 3; for n > 3, a(n) is the smallest unused positive number that shares a factor with the largest odd divisor of a(n-1).

%I A373698 #19 Jun 25 2024 18:04:06
%S A373698 1,2,3,6,9,12,15,5,10,20,25,30,18,21,7,14,28,35,40,45,24,27,33,11,22,
%T A373698 44,55,50,60,36,39,13,26,52,65,70,42,48,51,17,34,68,85,75,54,57,19,38,
%U A373698 76,95,80,90,63,49,56,77,66,69,23,46,92,115,100,105,72,78,81,84,87,29,58,116,145,110
%N A373698 a(n) = n for n <= 3; for n > 3, a(n) is the smallest unused positive number that shares a factor with the largest odd divisor of a(n-1).
%C A373698 The terms studied are all concentrated along lines of different gradient, the prime numbers, which appear in their natural order, forming the lowermost line. The central line has the highest concentration of terms and is composed of numbers with various numbers of prime factors, while many of the other lines consist of terms with the same number of prime factors. The odd terms all fall into the highest, lowest, and middle line, all other lines containing only even terms. It is conjectured the above behaviors are true for all n. See the attached images.
%C A373698 It appears that when a prime p > 3 is a term, it is immediately preceded by a term 3*p and followed by a term 2*p. This is true for at least the first 500000 terms. This pattern is similar to that seen in the EKG sequence A064413 but here the order of appearance of 2*p and 3*p is reversed.
%C A373698 Other than the first three terms the first fixed points are 42544 and 47312, and no more exist in the first 500000 terms. An examination of the graphs shows these points are due to one of the lines of concentration crossing the line a(n) = n, implying it, and probably most of the other lines, has a slight downward curvature.
%C A373698 In the first 100000 terms the longest run of consecutive odd and even terms is eight, starting n = 96230 and n = 30466 respectively. It in unknown if these runs can be of arbitrary length. No power of 2, other than 2 itself, can be a term.
%H A373698 Scott R. Shannon, <a href="/A373698/b373698.txt">Table of n, a(n) for n = 1..10000</a>
%H A373698 Michael De Vlieger, <a href="/A373698/a373698_2.png">Log log scatterplot of a(n)</a>, 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 and purple, where the latter additionally represents powerful numbers that are not prime powers.
%H A373698 Scott R. Shannon, <a href="/A373698/a373698.png">Image of the first 100000 terms</a>. Numbers with one, two, three, four, or five and more prime factors, counted with multiplicity, are shown as red, yellow, green, blue and violet respectively. The white line is a(n) = n.
%H A373698 Scott R. Shannon, <a href="/A373698/a373698_1.png">Image of the first 100000 terms</a>. The even terms are shown in yellow, the odd terms in green.
%e A373698 a(11) = 25 as a(10) = 20 which has a largest odd divisor of 5, and 25 is the smallest unused positive number that shares a factor with 5.
%t A373698 j = 3; nn = 120; c[_] := False; m[_] := 1;
%t A373698 Array[Set[{a[#], c[#]}, {#, True}] &, j]; u = j + 1;
%t A373698 Do[If[PrimePowerQ[j],
%t A373698     (k = m[#]; While[c[k #], k++]; k *= #; While[c[#  m[#]], m[#]++]) &@
%t A373698       FactorInteger[j][[1, 1]],
%t A373698      k = u; While[Or[c[k], CoprimeQ[j, k]], k++] ];
%t A373698   Set[{a[n], c[k], j}, {#, True, #/2^IntegerExponent[#, 2]}] &[k],
%t A373698   {n, j + 1, nn}];
%t A373698 Array[a, nn] (* _Michael De Vlieger_, Jun 20 2024 *)
%Y A373698 Cf. A373623, A373545, A373546, A064413, A005843, A005408.
%K A373698 nonn
%O A373698 1,2
%A A373698 _Scott R. Shannon_, Jun 13 2024