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 A296423 Lexicographically earliest sequence of distinct positive terms such that, among any two consecutive terms, we have a p-smooth number and a (p+1)-rough number for some p > 0. 2
 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 13, 12, 17, 14, 19, 15, 16, 21, 23, 18, 25, 24, 29, 20, 31, 22, 37, 26, 41, 27, 32, 33, 43, 28, 47, 30, 49, 36, 35, 48, 53, 34, 59, 38, 61, 39, 64, 45, 67, 40, 71, 42, 73, 44, 79, 46, 83, 50, 77, 54, 55, 72, 65, 81, 85, 89 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Equivalently, for any n > 0, min(gpf(a(n)), gpf(a(n+1))) < max(lpf(a(n)), lpf(a(n+1))), where gpf = A006530 and lpf = A020639. Also, for any n > 0, { a(n), a(n+1) } = { u, v } such that for any prime p and q, if p divides u and q divides v then p < q. A p-smooth number is only divisible by prime numbers <= p, whereas a p-rough number is only divisible by prime numbers >= p. This sequence is a permutation of the positive numbers, with inverse A296424: - we can always extend the sequence with the least prime number that does not divide the product of the earlier terms, - hence every prime number appear in the sequence, in increasing order, - for any v > 0, v can appear after any prime number > v; as there are infinitely many such prime numbers, v will eventually appear. The first known fixed points are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 33, 108, 192, 1250. Two consecutive terms are always coprime. LINKS Rémy Sigrist, Table of n, a(n) for n = 1..10000 Rémy Sigrist, PARI program for A296423 Rémy Sigrist, Scatterplot of the first 50000 terms Rémy Sigrist, Colored scatterplot of the first 10000 terms EXAMPLE The first terms, alongside their distinct prime factors, are:   n   a(n)  distinct prime factors   --  ----  ----------------------    1   1    none    2   2    2    3   3    3    4   4    2    5   5    5    6   6    2, 3    7   7    7    8   8    2    9   9    3   10  11    11   11  10    2, 5   12  13    13   13  12    2, 3   14  17    17   15  14    2, 7   16  19    19   17  15    3, 5   18  16    2   19  21    3, 7   20  23    23 PROG (PARI) See Links section. CROSSREFS Cf. A006530, A020639, A296424 (inverse). Sequence in context: A113218 A296424 A065708 * A065650 A065649 A262037 Adjacent sequences:  A296420 A296421 A296422 * A296424 A296425 A296426 KEYWORD nonn AUTHOR Rémy Sigrist, Dec 12 2017 STATUS approved

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Last modified April 8 07:53 EDT 2020. Contains 333313 sequences. (Running on oeis4.)