A375110 -id:A375110 - cp's OEIS frontend

A375112 Triangle T(n, k), n > 0, k = 0..n-1, read by rows; T(n, k) is the unique m such that A375110(m+1) = n and A375110(m) mod A375110(m+1) = k.

1, 3, 2, 5, 7, 4, 10, 9, 14, 6, 12, 16, 18, 8, 11, 21, 23, 20, 30, 15, 13, 26, 25, 32, 36, 40, 17, 22, 28, 34, 38, 42, 44, 19, 24, 27, 47, 49, 51, 53, 46, 72, 31, 33, 29, 56, 58, 60, 55, 74, 82, 90, 37, 35, 48, 63, 65, 62, 76, 84, 92, 98, 41, 39, 50, 57, 68, 67, 78, 86, 94, 100, 104, 108, 43, 52, 59, 64

Offset: 1

Rémy Sigrist, Jul 30 2024

As a flat sequence, this is a permutation of the positive integers with inverse A375113.

			Triangle T(n, k) begins:
    1;
    3, 2;
    5, 7, 4;
    10, 9, 14, 6;
    12, 16, 18, 8, 11;
    21, 23, 20, 30, 15, 13;
    26, 25, 32, 36, 40, 17, 22;
    28, 34, 38, 42, 44, 19, 24, 27;
    47, 49, 51, 53, 46, 72, 31, 33, 29;
    ...

Rémy Sigrist, Table of n, a(n) for n = 1..10011 (rows for n = 1..141 flattened)
Rémy Sigrist, Colored representation of the first 1000 rows
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A375110, A375113 (inverse).

PARI
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A375111 a(n) = A375110(n) mod A375110(n+1).

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 1, 3, 1, 0, 4, 0, 5, 2, 4, 1, 5, 2, 5, 2, 0, 6, 1, 6, 1, 0, 7, 0, 8, 3, 6, 2, 7, 1, 8, 3, 7, 2, 8, 4, 7, 3, 8, 4, 8, 4, 0, 9, 1, 9, 2, 9, 3, 9, 3, 0, 10, 1, 10, 2, 10, 2, 0, 11, 1, 11, 1, 0, 12, 0, 13, 5, 9, 4, 10, 3, 11, 2, 12, 1, 13, 5, 10

Offset: 1

Rémy Sigrist, Jul 30 2024

nonn

			a(42) = A375110(42) mod A375110(43) = 11 mod 8 = 3.

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

Cf. A375110.

PARI
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A375114 Lexicographically earliest sequence of positive integers such that the pairs (gcd(a(n), a(n+1)), a(n+1)) are all distinct.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 4, 6, 6, 7, 7, 8, 8, 9, 6, 8, 10, 10, 11, 10, 12, 8, 12, 9, 9, 12, 12, 13, 12, 14, 14, 15, 10, 15, 14, 16, 16, 17, 16, 18, 12, 15, 15, 18, 18, 19, 18, 20, 16, 20, 20, 21, 14, 21, 20, 22, 11, 22, 22, 23, 22, 24, 16, 24, 18, 21

Offset: 1

Rémy Sigrist, Jul 30 2024

nonn

The value 1 appears twice; for any v > 1, the value v appears A000005(v) times.

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

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

See A375110 for similar sequences.

Cf. A000005.

PARI
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A375115 Lexicographically earliest sequence of positive integers such that the pairs (a(n) AND a(n+1), a(n+1)) are all distinct (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 3, 5, 3, 6, 4, 5, 5, 6, 6, 7, 7, 8, 5, 7, 9, 6, 9, 7, 10, 7, 11, 7, 12, 7, 13, 8, 7, 14, 9, 9, 10, 10, 11, 11, 12, 11, 13, 11, 14, 12, 13, 13, 14, 14, 15, 15, 16, 10, 13, 15, 17, 11, 15, 18, 11, 19, 12, 15, 20, 11, 20, 13, 18, 13, 19

Offset: 0

Rémy Sigrist, Jul 30 2024

The value 0 appears twice; for any v > 0, the value v appears A001316(v) times.

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

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program

See A375110 for similar sequences.

Cf. A001316.

PARI
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cp's OEIS Frontend

A375112 Triangle T(n, k), n > 0, k = 0..n-1, read by rows; T(n, k) is the unique m such that A375110(m+1) = n and A375110(m) mod A375110(m+1) = k.

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A375111 a(n) = A375110(n) mod A375110(n+1).

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A375114 Lexicographically earliest sequence of positive integers such that the pairs (gcd(a(n), a(n+1)), a(n+1)) are all distinct.

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PARI

A375115 Lexicographically earliest sequence of positive integers such that the pairs (a(n) AND a(n+1), a(n+1)) are all distinct (where AND denotes the bitwise AND operator).

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Author

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Examples

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Crossrefs

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PARI