A365117 a(1) = 1. Thereafter a(n) is the least novel multiple m of the smallest prime which does not divide a(n-1) and such that m is coprime to a(n-1).

1, 2, 3, 4, 9, 8, 15, 14, 27, 10, 21, 16, 33, 20, 39, 22, 45, 26, 51, 28, 57, 32, 63, 34, 69, 38, 75, 44, 81, 40, 87, 46, 93, 50, 99, 52, 105, 58, 111, 56, 117, 62, 123, 64, 129, 68, 135, 74, 141, 70, 153, 76, 147, 80, 159, 82, 165, 86, 171, 88, 177, 92, 183, 94

Offset: 1

Author

David James Sycamore, Aug 22 2023

Keywords

• nonn

Comments

The Name is as for A351495, but with an extra constraint: namely that a(n) is prime to a(n-1).

{a(n)}; n > 1 is a permutation of A047228.

Examples

			a(2) = 2 because 2 is the smallest prime which does not divide 1, and 2 is prime to 1.
3(3) = 3 because 3 is the smallest prime which does not divide 2, and 3 is prime to 2.
a(4) = 4 since it is the second multiple of 2, the smallest prime that does not divide 3, and 4 is prime to 3.
a(5) = 9 since it is the least novel multiple of 3, the smallest prime that does not divide 4, and is the least such number prime to 4.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..256, showing a(n) mod 6 = 3 in red, a(n) mod 6 = 2 in large dark blue, and a(n) mod 6 = 4 in green dots. Demonstrates 2 trajectories, an early trajectory pertaining to 3 (mod 6) and a late trajectory pertaining to +-2 (mod 6).
• Michael De Vlieger, 32 X 32 table of a(n), n = 1..1024, listed in rows, with a color code according to residue (mod 6). Blue = 2 (mod 6), red = 3 (mod 6), light green = 4 (mod 6), black = 0, 1, or 5 mod 6.
• Michael De Vlieger, 30 X 30 table of a(n), n = 1..900, listed in rows, with a color code as follows: black = empty product, red = prime, gold = composite prime power, light green = squarefree semiprime, green = squarefree number with more than 2 prime factors, purple = powerful number (in A001694) that is not a prime power, blue = other numbers that are neither squarefree nor prime powers.
• Michael De Vlieger, 210 X 210 table of a(n), n = 1..44100, listed in rows with color code as immediately above, demonstrating the aggregate pattern demonstrated in the sequence. Certain residues (mod 210) seem to be devoid of squarefree numbers, perhaps since they are a product 9*m.

Crossrefs

Cf. A047228, A351495.

Programs

Mathematica

nn = 12; m[] := 1; a[1] = j = 1; m[1] = 2; c[1] = True; c[] := False; f[x_] := Block[{q}, q = 2; While[! CoprimeQ[q, x], q = NextPrime[q]]; q]; Do[{k = m[#]; While[Or[! CoprimeQ[j, k], c[# k]], k++]; If[k == m[#], While[c[m[#] #], m[#]++]]; Set[{a[n], j, c[k #]}, {k #, k #, True}]} &@ f[j], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Aug 22 2023 *)

PARI

first(n) = {my(res = vector(n)); sofar = Set([1..4]); for(i = 1, 4, res[i] = i); for(i = 5, n, res[i] = nxt(res[i-1])); res}
nxt(n) = {my(start, step); if(n % 2 == 0, start = 3; step = 6, start = 2; step = [2,4]); forstep(i = start, oo, step, s = Set(i); if(gcd(i, n) == 1 && #setminus(s, sofar) == 1, sofar = setunion(sofar, s); return(i)))} \\ David A. Corneth, Aug 22 2023

Formula

From Michael De Vlieger, Aug 22 2023: (Start)

a(2n) == +-2 (mod 6).

a(2n+1) == 3 (mod 6), n > 0. (End)

Extensions

More terms from David A. Corneth, Aug 22 2023