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A282291 Lexicographically earliest sequence of distinct terms such that every pair of consecutive terms contains a term that is a unitary divisor of the other term.

%I A282291 #37 Jul 18 2022 22:47:13
%S A282291 1,2,6,3,12,4,20,5,10,30,15,60,420,7,14,42,21,84,28,140,35,70,210,105,
%T A282291 840,8,24,120,40,280,56,168,1848,11,22,66,33,132,44,220,55,110,330,
%U A282291 165,660,4620,77,154,462,231,924,308,1540,385,770,2310,1155,9240,88
%N A282291 Lexicographically earliest sequence of distinct terms such that every pair of consecutive terms contains a term that is a unitary divisor of the other term.
%C A282291 This sequence has connections with A113552 and A281978: each pair of consecutive terms contains a term that divides the other term.
%C A282291 The derived sequence A282304 gives some insights about the fractal nature of this sequence.
%C A282291 Conjectures:
%C A282291 - All prime numbers appear in this sequence, in increasing order,
%C A282291 - the derived sequence A282304 is unbounded,
%C A282291 - this sequence is a permutation of the natural numbers.
%C A282291 From _Antti Karttunen_, May 17 2018: (Start)
%C A282291 The greedy algorithm which constructs this sequence can be understood also in terms of Heinz encodings of partitions (see A215366): Any term a(n) corresponds to a particular integer partition {s1+...+sk} via mapping a(n) = prime(s1)*...*prime(sk), where s1 .. sk are the summands of an integer partition. The choices for constructing the next partition are: either remove some parts from the partition, but with the constraint that if any summand k is removed, then all copies of k present in partition must be removed in too. One may remove all copies of several distinct summands as well. If by such a removal of parts we can find any smaller partitions that have not yet occurred in the sequence, then we choose the one which has the smallest Heinz encoding value to be a(n+1). On the other hand, if all partitions obtained by such removals have already occurred in the sequence, one must then add one or more parts to the current partition, but with the constraint that one is allowed to use only summands that do not already occur in partition (but any number of such summands may be used, also of more than one kind, as long as such summands are not already present in the partition that corresponds to a(n)). Of all such valid new partitions not already encountered, one with the smallest Heinz encoding value is chosen to be a(n+1). Compare this to the rules given for similar A304531 and A303751.
%C A282291 Primes 2 .. 61 occur at: 2, 4, 8, 14, 34, 96, 193, 386, 770, 1538, 3074, 14647, 30533, 60824, 122349, 245225, 688293, 1535694.
%C A282291 Terms just before primes are: 1, 6, 20, 420, 1848, 6552, 556920, 1511640, 6953544, 11090902680, 26447537160, 444799488600, 411767273946600, 1361999444592600, 448097817270965400, 2159016755941924200, 768250528363503385200, 3827047701385526108400.
%C A282291 Primorials (A002110) occur at: 1, 2, 3, 10, 23, 56, 151, 343, 728, 1497, 3034, 6107, 20753, 51285, 112674, 235085, 655721, 1525973, 3151033, ...
%C A282291 Powers of 2: 2 .. 32 occur at: 2, 6, 26, 6531, 1210614, and immediately following terms are: 6, 20, 24, 48, 96.
%C A282291 Immediately preceding terms are: 1, 12, 840, 1163962800, 1479723952477818247200. After 1 these factor as: (2^2 * 3^1), (2^3 * 3^1 * 5^1 * 7^1), (2^4 * 3^2 * 5^2 * 7^1 * 11^1 * 13^1 * 17^1 * 19^1), (2^5 * 3^2 * 5^2 * 7^2 * 11^1 * 13^1 * 17^1 * 19^1 * 23^1 * 29^1 * 31^1 * 41^1 * 43^1 * 47^1 * 53^1).
%C A282291 Observed recurrences: From n>=4 and k>=2 onward, there is a following general pattern:
%C A282291 For n = x .. x+(y-1), a(n) = prime(1+k)*a(n-(x-1)),
%C A282291 where y is the k-th record in A282304, and x is the position of that record in A282304, starting from the k = 2nd record in that sequence:
%C A282291 For n = 8 .. 8+4, a(n) = 5*a(n-7).
%C A282291 For n = 14 .. 14+10, a(n) = 7*a(n-13).
%C A282291 For n = 34 .. 34+30, a(n) = 11*a(n-33).
%C A282291 For n = 96 .. 96+89, a(n) = 13*a(n-95).
%C A282291 For n = 193 .. 193+184, a(n) = 17*a(n-192).
%C A282291 For n = 386 .. 386+382, a(n) = 19*a(n-385).
%C A282291 For n = 770 .. 770+766, a(n) = 23*a(n-769).
%C A282291 For n = 1538 .. 1538+1534, a(n) = 29*a(n-1537).
%C A282291 For n = 3074 .. 3074+3070, a(n) = 31*a(n-3073).
%C A282291 For n = 14647 .. 14647+11104, a(n) = 37*a(n-14646).
%C A282291 For n = 30533 .. 30533+29454, a(n) = 41*a(n-30532).
%C A282291 For n = 60824 .. 60824+30061, a(n) = 43*a(n-60823).
%C A282291 For n = 122349 .. 122349+91330, a(n) = 47*a(n-122348).
%C A282291 For n = 245225 .. 245225+121950, a(n) = 53*a(n-245224).
%C A282291 For n = 688293 .. 688293+367237, a(n) = 59*a(n-688292).
%C A282291 For n = 1535694 .. 1535694+596154, a(n) = 61*a(n-1535693).
%C A282291 Note how this forces values like prime powers to gaps between. E.g. 49 = a(367278) occurs 103 steps after the subsection a(n) = 53*a(n-245224) has ended at 245225+121950 (= 367175), but before the next regular subsection a(n) = 59*a(n-688292) starts at 688293.
%C A282291 (End)
%H A282291 Rémy Sigrist, <a href="/A282291/b282291.txt">Table of n, a(n) for n = 1..10000</a>
%H A282291 Rémy Sigrist, <a href="/A282291/a282291.png">Logarithmic scatterplot of the first 250000 terms</a> (the two blue sections are equal up to a scaling factor of 47)
%H A282291 Rémy Sigrist, <a href="/A282291/a282291.gp.txt">PARI program for A282291</a>
%F A282291 For all n >= 1, A052331(a(n)) = A302853(n-1), A001222(a(n)) = A304099(n). - _Antti Karttunen_, May 17 2018
%e A282291 The first terms, alongside their p-adic valuations with respect to p=2, 3, 5 and 7 (with 0's omitted), are:
%e A282291 n  a(n)  v2 v3 v5 v7
%e A282291 -- ----  -- -- -- --
%e A282291 1     1
%e A282291 2     2   1
%e A282291 3     6   1  1
%e A282291 4     3      1
%e A282291 5    12   2  1
%e A282291 6     4   2
%e A282291 7    20   2     1
%e A282291 8     5         1
%e A282291 9    10   1     1
%e A282291 10   30   1  1  1
%e A282291 11   15      1  1
%e A282291 12   60   2  1  1
%e A282291 13  420   2  1  1  1
%e A282291 14    7            1
%e A282291 15   14   1        1
%e A282291 16   42   1  1     1
%e A282291 17   21      1     1
%e A282291 18   84   2  1     1
%e A282291 19   28   2        1
%e A282291 20  140   2     1  1
%e A282291 21   35         1  1
%e A282291 22   70   1     1  1
%e A282291 23  210   1  1  1  1
%e A282291 24  105      1  1  1
%e A282291 25  840   3  1  1  1
%t A282291 a = {1}; Do[k = 1; While[Or[MemberQ[a, k], Nand[Divisible[#2, #1], CoprimeQ[#1, #2/#1]]] & @@ Sort@ # &@{k, Last@ a}, k++]; AppendTo[a, k], {n, 58}]; a (* _Michael De Vlieger_, Feb 12 2017 *)
%o A282291 (PARI)
%o A282291 up_to = 2^23;
%o A282291 v282291 = vector(up_to);
%o A282291 m304090 = Map();
%o A282291 prev=1; for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m304090,d) && (1==gcd(d, prev/d)),v282291[n] = d;mapput(m304090,d,n);break)); if(!v282291[n], m = 2; try = m*prev; while(mapisdefined(m304090,try) || (gcd(prev, try/prev)!=1), m++; try = m*prev); v282291[n] = try; mapput(m304090,try,n)); prev = v282291[n]);
%o A282291 A282291(n) = v282291[n];
%o A282291 A304090(n) = mapget(m304090,n); \\ _Antti Karttunen_, May 17 2018
%Y A282291 Cf. A113552, A281978, A282304, A304531, A302853, A304099.
%Y A282291 Cf. A304097, A304098 (see the scatter plots for alternative perspectives).
%Y A282291 Cf. A304090 (inverse).
%K A282291 nonn
%O A282291 1,2
%A A282291 _Rémy Sigrist_, Feb 11 2017