This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A304531 #44 Sep 29 2018 18:48:46
%S A304531 1,2,6,3,12,4,36,9,18,90,5,10,30,15,60,20,180,45,360,8,24,120,40,1080,
%T A304531 27,54,270,135,540,108,2700,25,50,150,75,300,100,900,225,450,3150,7,
%U A304531 14,42,21,84,28,252,63,126,630,35,70,210,105,420,140,1260,315,2520,56,168,840,280,7560,189,378,1890,945,3780,756,18900,175,350,1050,525,2100,700
%N A304531 Suspected divisor-or-multiple permutation: a(1) = 1, and for n > 1, a(n) is either the least unitary divisor of a(n-1) not already present, or (if all unitary divisors already used), a(n) = a(n-1) * {the least power of the least prime not dividing a(n-1) such that the term is not already present}.
%C A304531 The greedy algorithm which constructs the sequence is easiest to grasp in terms of Heinz encodings of partitions (see A215366): Any term a(n) corresponds to a particular integer partition. The choices for constructing the next partition are: either remove some parts from the partition, but with the condition that if any summand k is removed, then all copies of k present in partition must be removed in toto. 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. On the other hand, if all partitions obtained by such removals have already occurred in the sequence, then one adds to the current partition the least number of copies of the least positive integer that is not yet a part of the partition (see A257993), until a partition is found which is not yet in the sequence. This process also implies that one never removes the summand(s) that was/were just added in the previous step.
%C A304531 It has not yet been rigorously proved that all partitions can be reached this way, i.e., that this sequence is a permutation of natural numbers.
%C A304531 Each a(n+1) is always either a divisor or a multiple of a(n).
%C A304531 No two successive descending terms, that is, a(n) > a(n+1) > a(n+2) never occurs.
%C A304531 For n > 1, if a(n) is odd then a(n-1) = 2^h * k * a(n) and a(n+1) = 2^j * a(n) for some h, k and j, that is, odd terms occur between two larger even numbers.
%C A304531 If a(n) < a(n+1) then (a(n+1) / a(n)) is a divisor of a(n+2). This follows because clearly (in case A) when a(n) < a(n+1) < a(n+2) then (a(n+1) / a(n)) is a divisor of a(n+2) because on ascending subsections each successive term is obtained by multiplying by some prime (or its power) not already present. But it is also true (in case B) when a(n) < a(n+1) > a(n+2), as:
%C A304531 In contrast to A303751, this permutation is specified with an additional constraint that gcd(a(n+1), a(n)/a(n+1)) = 1, whenever a(n) > a(n+1). From this then follows that also when a(n) < a(n+1) > a(n+2) then (a(n+1) / a(n)) is guaranteed to be a divisor of a(n+2). It also follows from this that also the squarefree version A304537(n) = A019565(A052331(a(1+n))) satisfies the divisor-or-multiple property.
%C A304531 Odd numbers occur at A304530.
%C A304531 Primes occur at : 2, 4, 11, 42, 237, 1798, 7192, 69611, 431203, 2401568, ...
%C A304531 Primorials (A002110) occur at: 1, 2, 3, 13, 54, 290, 2087, 11333, 118777, 934737, ...
%H A304531 Antti Karttunen, <a href="/A304531/b304531.txt">Table of n, a(n) for n = 1..8447</a>
%H A304531 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A304531 Observed patterns:
%F A304531 For n = 2 .. 2+0, a(n) = 2*a(n-1).
%F A304531 For n = 4 .. 4+0, a(n) = 3*a(n-3).
%F A304531 For n = 11 .. 11+7, a(n) = 5*a(n-10).
%F A304531 For n = 42 .. 42+38, a(n) = 7*a(n-41).
%F A304531 For n = 237 .. 237+64, a(n) = 11*a(n-236).
%F A304531 For n = 1798 .. 1798+336, a(n) = 13*a(n-1797).
%F A304531 For n = 7192 .. 7192+1255, a(n) = 17*a(n-7191).
%F A304531 For n = 69611 .. 69611+4820, a(n) = 19*a(n-69610).
%F A304531 For n = 431203 .. 431203+41802, a(n) = 23*a(n-431202).
%F A304531 For n = 2401568 .. 2401568+131366, a(n) = 29*a(n-2401567).
%F A304531 Derived sequences. For all n >= 1:
%F A304531 A052331(a(n)) = A304533(n-1).
%F A304531 A064547(a(n)) = A304536(n-1).
%e A304531 a(64) = 280 = 2^3 * 5 * 7 = prime(1)^3 * prime(3) * prime(4), which by Heinz-encoding corresponds to integer partition {1+1+1+3+4}. We try to remove all 1's (to get {3+4}, i.e. prime(3)*prime(4) = 35, but that has already been used as a(52)), or to remove either 3 or 4 or both, but also 8, 40 and 56 have already been used, and if we remove all 1's and either 3 or 4, then also prime(3) and prime(4), 5 and 7 have already been used. So we must add one or more copies of 2 (the least missing part) to find a partition that has not already been used. And it turns out we need to add three copies, to get {1+1+1+2+2+2+3+4} to obtain value prime(1)^3 * prime(2)^3 * prime(3) * prime(4) = 7560 not used before, so a(65) = 7560.
%e A304531 For the next partition, we remove all 1's and the sole 3, to get {2+2+2+4}, with Heinz-encoding prime(2)^3 * prime(4) = 27 * 7 = 189 to obtain the smallest not yet present in sequence, thus a(66) = 189. Note that the partition {1+1+1+2+2} would give even a smaller Heinz-code 2^3 * 3^2 = 72, which also has not been used before, but 72 is not a unitary divisor of 7560, which can be seen from the fact that just one 2 (but not all 2's) was removed from the partition {1+1+1+2+2+2+3+4} that corresponds to 7560. At this point A303751 selects 72 as it has no unitary divisor constraint.
%o A304531 (PARI)
%o A304531 up_to = 2^12;
%o A304531 A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
%o A304531 v304531 = vector(up_to);
%o A304531 m304532 = Map();
%o A304531 prev=1; for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m304532,d) && (1==gcd(d, prev/d)),v304531[n] = d;mapput(m304532,d,n);break)); if(!v304531[n], p = A053669(prev); while(mapisdefined(m304532,prev), prev *= p); v304531[n] = prev; mapput(m304532,prev,n)); prev = v304531[n]);
%o A304531 A304531(n) = v304531[n];
%o A304531 A304532(n) = mapget(m304532,n);
%Y A304531 Cf. A304532 (inverse).
%Y A304531 Cf. A304530 (positions of odd terms).
%Y A304531 Cf. A053669, A215366, A257993, A304533, A304535, A304536, A304537.
%Y A304531 Cf. A113552, A282291, A303751 for other variants.
%Y A304531 Differs from A303751 for the first time at n=66, where a(66) = 189, while A303751(66) = 72.
%K A304531 nonn
%O A304531 1,2
%A A304531 _Antti Karttunen_, May 14 2018