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.

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, 11, 33, 30, 44, 36, 40, 55, 66, 77, 63, 88, 70, 45, 99, 110, 121, 39, 13, 26, 48, 52, 32, 51, 17, 34, 54, 68, 38, 19, 57, 60, 76, 69, 23, 46, 72, 92, 50, 65, 115, 138, 161, 84, 184

Offset: 1

Author

David James Sycamore, Mar 02 2023

Keywords

• nonn
• look

Comments

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.

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).

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.

Conjectured to be a permutation of the natural numbers, with primes in natural order.

Examples

			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.
a(5) = 9 since h,j = 2,6 with difference 3, and 9 is the least novel multiple of 3.
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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20, showing cases resulting from rad(h) = rad(j) in red, else blue.
• Michael De Vlieger, Log log scatterplot of a(n), 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.
• Michael De Vlieger, Log log scatterplot of a(n), 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.

Crossrefs

Cf. A007947, A064413, A336957.

Programs

Mathematica

nn = 2^10; c[] = False; q[] = 1;
f[n_] := f[n] = FactorInteger[n][[All, 1]];
Array[Set[{a[#], c[#]}, {#, True}] &, 3];
Set[{h, i, j, R, S, T}, {a[1], a[2], a[3], f[a[1]], f[a[2]],
  f[a[3]]}]; u = 4;
Do[If[R == T,
   k = u; While[Or[c[k], CoprimeQ[i, k]], k++],
   (k = q[#]; While[c[k #], k++]; k *= #; While[c[# q[#]], q[#]++]) &[
    Max@ SymmetricDifference[R, T] ] ];
  Set[{a[n], c[k], h, i, j}, {k, True, i, j, k}];
  Set[{R, S, T}, {S, T, f[k]}];
  If[k == u, While[c[u], u++]], {n, 4, nn}];
Array[a, nn] (* Michael De Vlieger, Mar 05 2023 *)