A358277 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous Omega(a(n-1)) terms.

1, 2, 3, 4, 5, 6, 7, 8, 11, 9, 10, 13, 12, 17, 14, 15, 19, 16, 23, 18, 25, 29, 20, 21, 31, 22, 27, 35, 26, 33, 37, 24, 41, 28, 43, 30, 47, 32, 53, 34, 39, 49, 38, 45, 59, 36, 61, 40, 67, 42, 71, 44, 65, 51, 46, 55, 57, 52, 73, 48, 79, 50, 77, 69, 58, 83, 54, 85, 89, 56, 97, 60, 101, 62, 63, 95

Offset

1,2

Comments

Unlike A356850 all the terms are concentrated along three straight lines. In the first 100000 terms there are ten fixed points, 1, 2, 3, ..., 27, 57, and it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^10, with records in red, local minima in blue, highlighting primes in green and other prime powers in gold, showing numbers that are neither squarefree nor prime powers in magenta.

Michael De Vlieger, Scalar scatterplot of a(n), n = 1..2^12, showing primes in red, composite prime powers in gold, composite squarefree numbers in green, products of composite prime powers in magenta, and other numbers in blue.

Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^10 using a color function showing Omega(a(n-1)).

Example

a(4) = 4 as a(3) = 3 and Omega(3) = A001222(3) = 1, and 4 is coprime to the previous one term, namely a(3) = 3.
a(9) = 11 as a(8) = 8 and Omega(8) = A001222(8) = 3, and 11 is coprime to the previous three terms, namely 8, 7, 6.

Mathematica

nn = 90; c[_] = False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = 3; j = 2; Do[k = u; m = Times @@ Array[a[n - #] &, PrimeOmega[j]]; While[Nand[! c[k], CoprimeQ[k, m]], k++]; Set[{a[n], c[k], j}, {k, True, k}]; If[k == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Nov 20 2022 *)

Crossrefs

Cf. A356850, A356903, A001222, A356851, A093714, A336957, A000040.

Sequence in context: A280770 A354441 A354435 * A361513 A255004 A257728

Adjacent sequences: A358274 A358275 A358276 * A358278 A358279 A358280

Keyword

nonn

Author

Scott R. Shannon, Nov 08 2022

Status

approved