A335943 -id:A335943 - cp's OEIS frontend

A375110 Lexicographically earliest sequence of positive integers such that the pairs (a(n) mod a(n+1), a(n+1)) are all distinct.

1, 1, 2, 2, 3, 3, 4, 3, 5, 4, 4, 5, 5, 6, 4, 6, 5, 7, 5, 8, 6, 6, 7, 6, 8, 7, 7, 8, 8, 9, 6, 9, 7, 9, 8, 10, 7, 10, 8, 11, 7, 11, 8, 12, 8, 13, 9, 9, 10, 9, 11, 9, 12, 9, 13, 10, 10, 11, 10, 12, 10, 13, 11, 11, 12, 11, 13, 12, 12, 13, 13, 14, 9, 14, 10, 14, 11

Offset: 1

Rémy Sigrist, Jul 30 2024

nonn

The value 1 appears twice; for any v > 1, v appears v times.

This sequence shares graphical features with A335943.

			The first terms, alongside the corresponding pairs, are:
  n   a(n)  (a(n) mod a(n+1), a(n+1))
  --  ----  -------------------------
   1     1  (0, 1)
   2     1  (1, 2)
   3     2  (0, 2)
   4     2  (2, 3)
   5     3  (0, 3)
   6     3  (3, 4)
   7     4  (1, 3)
   8     3  (3, 5)
   9     5  (1, 4)
  10     4  (0, 4)
  11     4  (4, 5)
  12     5  (0, 5)
  13     5  (5, 6)
  14     6  (2, 4)
  15     4  (4, 6)

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

See A375114 and A375115 for similar sequences.

Cf. A335943, A375111, A375112.

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A335944 Lexicographically earliest sequence of positive integers such that for any distinct m and n, the fractional parts of m/a(m) and of n/a(n) are distinct.

Original entry on oeis.org

1, 3, 2, 3, 4, 5, 4, 5, 5, 7, 6, 5, 6, 9, 7, 7, 8, 7, 7, 7, 8, 9, 8, 11, 9, 9, 8, 9, 9, 11, 10, 11, 10, 11, 12, 11, 10, 11, 10, 11, 12, 11, 12, 13, 13, 13, 13, 11, 12, 11, 13, 15, 13, 13, 13, 13, 14, 15, 14, 17, 13, 13, 13, 15, 14, 17, 14, 15, 14, 17, 15, 17

Offset: 1

Rémy Sigrist, Jul 01 2020

nonn

For any k > 0, k appears A000010(k) times.

			The first terms, alongside the fractional part of n/a(n), are:
  n   a(n)  frac(n/a(n))
  --  ----  ------------
   1     1       0
   2     3      2/3
   3     2      1/2
   4     3      1/3
   5     4      1/4
   6     5      1/5
   7     4      3/4
   8     5      3/5
   9     5      4/5
  10     7      3/7

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of (n, frac(n/a(n))) for n = 1..50000

See A335943 for a similar sequence.

Cf. A000010.

ff = []; for (n=1, 72, for (v=1, oo, if (!setsearch(ff, f=frac(n/v)), print1 (v ", "); ff=setunion(ff, [f]); break)))

cp's OEIS Frontend

A375110 Lexicographically earliest sequence of positive integers such that the pairs (a(n) mod a(n+1), a(n+1)) are all distinct.

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