A338441 -id:A338441 - cp's OEIS frontend

A338338 Lexicographically earliest infinite sequence of distinct positive numbers such that for any prime p, any run of consecutive multiples of p has length exactly 3.

1, 2, 4, 6, 3, 9, 5, 10, 20, 8, 7, 14, 28, 12, 15, 30, 40, 16, 11, 22, 44, 18, 21, 42, 56, 24, 27, 33, 55, 110, 50, 26, 13, 39, 36, 48, 32, 17, 34, 68, 38, 19, 57, 45, 60, 70, 84, 63, 51, 85, 170, 80, 46, 23, 69, 54, 66, 88, 77, 35, 105, 75, 72, 52, 78, 117

Offset: 1

N. J. A. Sloane, Oct 27 2020

nonn

If a prime p divides a(n), then there is a run of exactly three terms (one of which is a(n)) that are divisible by p.
If "three" is changed to "two", we get A280864.
Conjecture: This is a permutation of the positive integers.

			After 1,2,4,6,3, we have had two successive multiples of 3, so the next term must be a multiple of 3 we have not yet seen, hence 9. The following term is then the smallest number not yet seen which is not a multiple of 3, hence 5.

Rémy Sigrist, Table of n, a(n) for n = 1..20000
Rémy Sigrist, PARI program for A338338

A338339-A338349, A338440, A338449, A338450, and A338451 analyze this sequence from various points of view.

Cf. A280864, A338441.

PARI
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Corrected and extended by Rémy Sigrist, Oct 27 2020

A338444 Lexicographically earliest sequence of distinct squarefree numbers such that for any prime p, any run of consecutive multiples of p has length exactly 2.

Original entry on oeis.org

1, 2, 6, 3, 5, 10, 14, 7, 11, 22, 26, 13, 15, 30, 34, 17, 19, 38, 42, 21, 23, 46, 58, 29, 31, 62, 66, 33, 35, 70, 74, 37, 39, 78, 82, 41, 43, 86, 94, 47, 51, 102, 106, 53, 55, 110, 114, 57, 59, 118, 122, 61, 65, 130, 134, 67, 69, 138, 142, 71, 73, 146, 154, 77

Offset: 1

Rémy Sigrist, Oct 28 2020

nonn

This sequence is a squarefree variant of A280864.

Conjecture: this sequence is a permutation of the squarefree numbers (A005117).

			The first terms, alongside their prime factors, are:
  n   a(n)  Prime factors
  --  ----  -------------
   1     1
   2     2  2
   3     6  2 3
   4     3    3
   5     5      5
   6    10  2   5
   7    14  2     7
   8     7        7
   9    11          11
  10    22  2       11
  11    26  2         13
  12    13            13
  13    15    3 5
  14    30  2 3 5
  15    34  2           17
  16    17              17

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A338444

Cf. A005117, A280864, A338441.

C
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A338338 Lexicographically earliest infinite sequence of distinct positive numbers such that for any prime p, any run of consecutive multiples of p has length exactly 3.

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