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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
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, 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 *)
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Nov 08 2022
STATUS
approved