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 A361133 #25 Nov 09 2024 04:39:49
%S A361133 1,2,3,6,9,4,12,8,10,5,15,18,20,21,7,14,24,28,16,27,35,42,49,25,56,22,
%T A361133 11,33,30,44,36,40,55,66,77,63,88,70,45,99,110,121,39,13,26,48,52,32,
%U A361133 51,17,34,54,68,38,19,57,60,76,69,23,46,72,92,50,65,115,138,161,84,184
%N A361133 a(n) = n for n <= 3. Let h, i, j represent a(n-3), a(n-2), a(n-1) respectively. For n > 3, if there is a symmetric difference in the sets of distinct primes dividing h and j, with greatest member p then a(n) is the least novel multiple of p. Otherwise, a(n) is the least novel k such that (k,i) > 1.
%C A361133 In other words if there are primes p which divide h but not j or primes q which divide j but not h then a(n) is the least novel multiple of the greatest of all such primes p, q. If there are no such primes (rad(h) = rad(j)), then a(n) is the least unused number sharing a divisor with i.
%C A361133 When an odd prime p appears it is immediately preceded and followed by multiples m*p and r*p of p respectively, thus m*p, p, r*p where if m = 2 then b is 3, and m > 2 forces r = 2 (compare with A064413 where m = 2, and r = 3 throughout).
%C A361133 The scatterplot resembles a fine-toothed comb, wherein it seems that the "teeth" represent consecutive multiples of certain distinct primes, which become compacted closer and closer together as the sequence progresses.
%C A361133 Conjectured to be a permutation of the natural numbers, with primes in natural order.
%H A361133 Michael De Vlieger, <a href="/A361133/b361133.txt">Table of n, a(n) for n = 1..16384</a>
%H A361133 Michael De Vlieger, <a href="/A361133/a361133.png">Log log scatterplot of a(n)</a>, n = 1..2^20, showing cases resulting from rad(h) = rad(j) in red, else blue.
%H A361133 Michael De Vlieger, <a href="/A361133/a361133_1.png">Log log scatterplot of a(n)</a>, n = 1..2^10, showing records in red, local minima in blue, terms resulting from rad(h) = rad(j) in green, and a dashed line showing a(n) = n.
%H A361133 Michael De Vlieger, <a href="/A361133/a361133_2.png">Log log scatterplot of a(n)</a>, n = 1..2^10, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither prime power nor squarefree in blue, highlighting products of distinct composite prime powers in light blue.
%e A361133 a(4) = 6 because the symmetric difference for 1 and 3 contains only one prime (3) and 6 is the least multiple of 3 that has not occurred already.
%e A361133 a(5) = 9 since h,j = 2,6 with difference 3, and 9 is the least novel multiple of 3.
%e A361133 a(6) = 4 since h,i,j = 3,6,9 (3,9 have no symmetric difference), and 4 is least novel number sharing a divisor with i = 6.
%t A361133 nn = 2^10; c[_] = False; q[_] = 1;
%t A361133 f[n_] := f[n] = FactorInteger[n][[All, 1]];
%t A361133 Array[Set[{a[#], c[#]}, {#, True}] &, 3];
%t A361133 Set[{h, i, j, R, S, T}, {a[1], a[2], a[3], f[a[1]], f[a[2]],
%t A361133 f[a[3]]}]; u = 4;
%t A361133 Do[If[R == T,
%t A361133 k = u; While[Or[c[k], CoprimeQ[i, k]], k++],
%t A361133 (k = q[#]; While[c[k #], k++]; k *= #; While[c[# q[#]], q[#]++]) &[
%t A361133 Max@ SymmetricDifference[R, T] ] ];
%t A361133 Set[{a[n], c[k], h, i, j}, {k, True, i, j, k}];
%t A361133 Set[{R, S, T}, {S, T, f[k]}];
%t A361133 If[k == u, While[c[u], u++]], {n, 4, nn}];
%t A361133 Array[a, nn] (* _Michael De Vlieger_, Mar 05 2023 *)
%Y A361133 Cf. A007947, A064413, A336957.
%K A361133 nonn,look
%O A361133 1,2
%A A361133 _David James Sycamore_, Mar 02 2023