A358026 Let G(n) = gcd(a(n-2),a(n-1)), a(1)=1, a(2)=2, a(3)=3. Thereafter if G(n) = 1, a(n) is the least novel m sharing a divisor with both a(n-2) and a(n-1). If G(n) > 1 and every prime divisor of a(n-1) also divides a(n-2), a(n) is the least m prime to both a(n-1) and a(n-2). Otherwise a(n) is the least novel multiple of any prime divisor of a(n-1) which does not divide a(n-2).

1, 2, 3, 6, 4, 5, 10, 8, 7, 14, 12, 9, 11, 33, 15, 20, 16, 13, 26, 18, 21, 28, 22, 44, 17, 34, 24, 27, 19, 57, 30, 25, 23, 115, 35, 42, 32, 29, 58, 36, 39, 52, 38, 76, 31, 62, 40, 45, 48, 46, 69, 51, 68, 50, 55, 66, 54, 37, 74, 56, 49, 41, 287, 63, 60, 64, 43

Offset

1,2

Comments

Conjectured to be a permutation of the positive integers with the primes in natural order, and primes are the slowest numbers to appear (as in A352187).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^14, showing records in red and local minima in blue, highlighting primes in green and other prime powers in gold.

Example

a(4) = 6, the least novel number sharing a factor with both 2 and 3.
a(5) = 4, the least novel multiple of 2, which divides a(4) but does not divide a(3).
Since every prime dividing a(5)=4 also divides a(4)=6, a(6)=5 the least novel term prime to 3 and 6.

Mathematica

nn = 67; c[_] = False; q[_] = 1; u = 4; Do[(Set[{a[n], c[n]}, {n, True}]; q[n]++), {n, u - 1}]; Do[m = FactorInteger[a[n - 1]][[All, 1]]; f = Select[m, CoprimeQ[#, a[n - 2]] &]; Which[Length[f] == PrimeNu[a[n - 1]], Set[{k, q[#1]}, {#2, #2/#1}] & @@ First@ MinimalBy[Map[{#, Set[g, q[#]]; While[c[g #], g++]; # g} &, Flatten@ Outer[Times, m, FactorInteger[a[n - 2]][[All, 1]] ] ], Last], Length[f] == 0, k = u; While[Nand[! c[k], CoprimeQ[a[n - 2], k], CoprimeQ[a[n - 1], k]], k++]; If[k == u, While[c[u], u++]], True, Set[{k, q[#1]}, {#2, #2/#1}] & @@ First@ MinimalBy[Map[{#, Set[g, q[#]]; While[c[g #], g++]; # g} &, f], Last] ]; Set[{a[n], c[k]}, {k, True}], {n, 4, nn}]; Array[a, nn] (* Michael De Vlieger, Oct 25 2022 *)

Crossrefs

Cf. A064413, A336957, A352187, A357963.

Sequence in context: A222758 A127915 A361966 * A331117 A282841 A254106

Adjacent sequences: A358023 A358024 A358025 * A358027 A358028 A358029

Keyword

nonn

Author

David James Sycamore, Oct 25 2022

Extensions

More terms from Michael De Vlieger, Oct 25 2022

Status

approved