Rémy Sigrist - cp's OEIS frontend

A387542 a(n) is the distance from the n-th term of A386482 to the nearest term of A386482 coprime to it.

0, 1, 2, 3, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 4, 2, 2, 2, 2, 3, 5, 4, 3, 5, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 3, 5, 5, 6, 5, 4, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 6, 8, 8, 9, 11

Offset: 1

Rémy Sigrist, Sep 01 2025

In other words: a(n) is the least d >= 0 such that gcd(A386482(n), A386482(n - d)) = 1 or gcd(A386482(n), A386482(n + d)) = 1.

The sequence is well defined as A386482(1) = 1 is coprime to all terms of A386482.

			For n = 7: the GCD of A386482(7) = 12 and its neighboring terms are:
  d   A387542(7+d)  gcd(A387542(7), A387542(7+d))
  --  ------------  -----------------------------
  -4             4                              4
  -3             6                              6
  -2             3                              3
  -1             9                              3
   0            12                             12
   1            10                              2
   2             8                              4
   3            14                              2
   4             7                              1
The nearest coprime term, A387542(11) = 7, is at distance 4, so a(7) = 4.

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

Cf. A386482.

PARI
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A387544 Even terms of A386482, halved, in order of occurrence.

Original entry on oeis.org

1, 2, 3, 6, 5, 4, 7, 9, 8, 10, 15, 14, 13, 12, 11, 18, 17, 16, 19, 27, 26, 25, 24, 23, 22, 21, 20, 30, 29, 28, 34, 33, 32, 31, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 46, 47, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50

Offset: 1

Rémy Sigrist, Sep 01 2025

Each record in the sequence is followed by the smaller missing terms in descending order.

			The first terms, alongside the first terms of A386482, are:
  k   A386482(k)  n   a(n)
  --  ----------  --  ----
   1           1
   2           2   1     1
   3           4   2     2
   4           6   3     3
   5           3
   6           9
   7          12   4     6
   8          10   5     5
   9           8   6     4
  10          14   7     7
  11           7
  12          21
  13          18   8     9
  14          16   9     8
  15          20  10    10

Rémy Sigrist, Table of n, a(n) for n = 1..10306
Rémy Sigrist, PARI program

Cf. A386482, A387558.

PARI
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a(a(n)) = n.

A387558 Numbers k such that A386482(k) is even.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 10, 13, 14, 15, 19, 20, 21, 22, 23, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 46, 47, 48, 53, 54, 55, 56, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 74, 75, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97

Offset: 1

Rémy Sigrist, Sep 02 2025

			A386482(46) = 60 is even, so 46 belongs to the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10306
Rémy Sigrist, PARI program

Cf. A386482, A387544.

PARI
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A387544(n) = A386482(a(n)) / 2.

A387094 An infinite variant of the EKG sequence (A064413) such that for any n > 0, a(2n) > a(2n-1) and a(2n) > a(2n+1).

Original entry on oeis.org

1, 4, 2, 6, 3, 12, 8, 10, 5, 15, 9, 18, 14, 20, 16, 22, 11, 33, 21, 27, 24, 26, 13, 39, 30, 32, 28, 34, 17, 51, 36, 38, 19, 57, 42, 44, 40, 45, 25, 35, 7, 56, 46, 50, 48, 54, 52, 58, 29, 87, 60, 62, 31, 93, 63, 66, 55, 70, 49, 84, 64, 72, 68, 74, 37, 111, 69

Offset: 1

Rémy Sigrist, Aug 16 2025

For the sequence to be infinite, when computing a term with even index, say a(2*n) for some n > 1, we must ensure that some missing value v < a(2*n) such that gcd(a(2*n), v) != 1 exists.

Will every integer appear in the sequence?

			Sequence begins:
  n   a(n)
  --  ----
   1     1
   2     4
   3     2
   4     6
   5     3
   6    12
   7     8
   8    10
   9     5
  10    15
  11     9
  12    18
  13    14
  14    20
  15    16
  16    22
  17    11

Rémy Sigrist, Table of n, a(n) for n = 1..10001
Rémy Sigrist, PARI program
Index entries for sequences related to EKG sequence

Cf. A064413.

PARI
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A387059 Inverse permutation to A387058.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 5, 3, 8, 10, 11, 13, 15, 16, 14, 12, 17, 9, 18, 20, 22, 24, 26, 28, 30, 31, 29, 27, 25, 23, 21, 19, 32, 34, 36, 38, 40, 41, 43, 45, 47, 48, 46, 44, 42, 49, 39, 37, 35, 33, 51, 53, 55, 57, 59, 50, 61, 63, 65, 67, 69, 70, 68, 66, 64, 62, 71, 60

Offset: 0

Rémy Sigrist, Aug 15 2025

nonn

			A387058(42) = 44, so a(44) = 42.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A387058.

PARI
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A387058 Lexicographically earliest sequence of distinct nonnegative integers such that each term is a square number or belongs to a run of two consecutive terms summing to a square number.

Original entry on oeis.org

0, 1, 2, 7, 3, 6, 4, 5, 8, 17, 9, 10, 15, 11, 14, 12, 13, 16, 18, 31, 19, 30, 20, 29, 21, 28, 22, 27, 23, 26, 24, 25, 32, 49, 33, 48, 34, 47, 35, 46, 36, 37, 44, 38, 43, 39, 42, 40, 41, 45, 55, 50, 71, 51, 70, 52, 69, 53, 68, 54, 67, 56, 65, 57, 64, 58, 63, 59

Offset: 0

Rémy Sigrist, Aug 15 2025

nonn

This sequence is a permutation of the nonnegative as each term belongs to a run of one or two terms summing to a square number, and after such a run we can extend the sequence with the least missing value.

			The first terms and corresponding square numbers are:
  n   a(n)  Squares
  --  ----  -----------------------------
   0     0  a(0) = 0^2
   1     1  a(0) + a(1) = 1^2, a(1) = 1^2
   2     2  a(2) + a(3) = 3^2
   3     7  a(2) + a(3) = 3^2
   4     3  a(4) + a(5) = 3^2
   5     6  a(4) + a(5) = 3^2
   6     4  a(6) = 2^2, a(6) + a(7) = 3^2
   7     5  a(6) + a(7) = 3^2
   8     8  a(8) + a(9) = 5^2
   9    17  a(8) + a(9) = 5^2
  10     9  a(10) = 3^2
  11    10  a(11) + a(12) = 5^2
  12    15  a(11) + a(12) = 5^2
  13    11  a(13) + a(14) = 5^2
  14    14  a(13) + a(14) = 5^2
  15    12  a(15) + a(16) = 5^2
  16    13  a(15) + a(16) = 5^2
  17    16  a(17) = 4^2

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A034175, A387059 (inverse).

PARI
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A387025 Start with the list of positive integers L_1 = (1, 2, ...); for n = 1, 2, ..., let m be the least integer > n such that L_n(n) divides L_n(m); L_{n+1}(k) = L_n(k) for any k <> m, L_{n+1}(m) = L_n(m)/L_n(n); a(n) = L_n(n).

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 8, 9, 2, 11, 6, 13, 2, 15, 1, 17, 2, 19, 10, 21, 2, 23, 2, 25, 1, 27, 28, 29, 2, 31, 16, 33, 2, 35, 18, 37, 2, 39, 2, 41, 1, 43, 44, 45, 2, 47, 3, 49, 1, 17, 52, 53, 2, 55, 1, 57, 2, 59, 30, 61, 2, 63, 32, 65, 2, 67, 2, 69, 1, 71, 4, 73, 2

Offset: 1

Rémy Sigrist, Aug 13 2025

nonn

Applying the same procedure to the powers of two yields A060546.

Applying the same procedure to the factorial numbers yields A006882.

			The first terms are:
  n   a(n)  L_n
  --  ----  ------------------------------------------------------
   1     1  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
   2     2  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
   3     3  1, 2, 3, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
   4     2  1, 2, 3, 2, 5, 2, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
   5     5  1, 2, 3, 2, 5, 1, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
   6     1  1, 2, 3, 2, 5, 1, 7, 8, 9,  2, 11, 12, 13, 14, 15, ...
   7     7  1, 2, 3, 2, 5, 1, 7, 8, 9,  2, 11, 12, 13, 14, 15, ...
   8     8  1, 2, 3, 2, 5, 1, 7, 8, 9,  2, 11, 12, 13,  2, 15, ...
   9     9  1, 2, 3, 2, 5, 1, 7, 8, 9,  2, 11, 12, 13,  2, 15, ...
  10     2  1, 2, 3, 2, 5, 1, 7, 8, 9,  2, 11, 12, 13,  2, 15, ...
  11    11  1, 2, 3, 2, 5, 1, 7, 8, 9,  2, 11,  6, 13,  2, 15, ...
  12     6  1, 2, 3, 2, 5, 1, 7, 8, 9,  2, 11,  6, 13,  2, 15, ...
  13    13  1, 2, 3, 2, 5, 1, 7, 8, 9,  2, 11,  6, 13,  2, 15, ...
  14     2  1, 2, 3, 2, 5, 1, 7, 8, 9,  2, 11,  6, 13,  2, 15, ...
  15    15  1, 2, 3, 2, 5, 1, 7, 8, 9,  2, 11,  6, 13,  2, 15, ...

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

Cf. A006882, A060546.

{ for (n = 1, #a = vector(74, n, n), print1 (a[n]", "); forstep (k = ceil((n+1)/a[n])*a[n], #a, a[n], if (a[k] % a[n]==0, a[k] /= a[n]; break;););); }

a(p) = p for any prime number p.

a(2*p) = 1 or 2 for any prime number p.

A387104 Split A386482 into maximal runs of consecutive decreasing terms; a(n) is the length of the n-th run.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 3, 3, 1, 6, 2, 3, 2, 10, 3, 5, 3, 5, 13, 3, 1, 2, 25, 2, 1, 3, 1, 1, 6, 7, 1, 3, 12, 4, 2, 33, 1, 1, 10, 6, 1, 11, 29, 51, 23, 10, 48, 61, 24, 26, 168, 1, 2, 2, 9, 1, 3, 2, 7, 2, 2, 6, 104, 15, 2, 1, 1, 2, 3, 3, 1, 1, 4, 11, 5, 159, 9, 1

Offset: 1

Rémy Sigrist, Aug 16 2025

nonn

			The first terms, alongside the corresponding runs, are:
  n   a(n)  Corresponding run
  --  ----  --------------------------------------
   1     1  1
   2     1  2
   3     1  4
   4     2  6, 3
   5     1  9
   6     3  12, 10, 8
   7     2  14, 7
   8     3  21, 18, 16
   9     3  20, 15, 5
  10     1  25
  11     6  30, 28, 26, 24, 22, 11
  12     2  33, 27
  13     3  36, 34, 32
  14     2  38, 19
  15    10  57, 54, 52, 50, 48, 46, 44, 42, 40, 35
  16     3  45, 39, 13

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

Cf. A386482, A387103.

PARI
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A387103(1 + Sum_{k = 1..n} a(k)) = 0 for any n > 0.

A387103 For any n >= 2, a(n) is the number of positive values k < A386482(n-1) missing from the first n-1 terms of A386482 such that gcd(k, A386482(n-1)) != 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 2, 0, 1, 0, 2, 2, 0, 2, 1, 0, 0, 5, 3, 3, 1, 1, 0, 1, 0, 2, 2, 0, 1, 0, 6, 10, 8, 7, 6, 4, 2, 3, 1, 0, 1, 1, 0, 2, 4, 2, 1, 0, 1, 1, 0, 3, 3, 1, 1, 0, 8, 16, 11, 9, 11, 7, 7, 6, 3, 3, 2, 1, 0, 1, 1, 0, 0, 1, 0, 17, 31, 22, 20, 28, 24, 16

Offset: 2

Rémy Sigrist, Aug 16 2025

nonn

This sequence gives essentially the number of candidates for A386482(n) that are less than A386482(n-1).

			The first terms, alongside A386482(n) and the corresponding k's, are:
  n   a(n)  A386482(n)  Candidates
  --  ----  ----------  --------------------
   1  N/A            1  N/A
   2     0           2  {}
   3     0           4  {}
   4     0           6  {}
   5     1           3  {3}
   6     0           9  {}
   7     0          12  {}
   8     2          10  {8, 10}
   9     2           8  {5, 8}
  10     0          14  {}
  11     1           7  {7}
  12     0          21  {}
  13     2          18  {15, 18}
  14     2          16  {15, 16}
  15     0          20  {}
  16     2          15  {5, 15}
  17     1           5  {5}
  18     0          25  {}
  19     0          30  {}
  20     5          28  {22, 24, 26, 27, 28}

Rémy Sigrist, Table of n, a(n) for n = 2..10000
Rémy Sigrist, Scatterplot of the candidates k for n = 2..1005
Rémy Sigrist, PARI program

Cf. A386482.

PARI
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a(n) = 0 iff A386482(n) > A386482(n-1).

A387087 GCD of pairs of consecutive terms of the sequence A386482.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 2, 2, 2, 7, 7, 3, 2, 4, 5, 5, 5, 5, 2, 2, 2, 2, 11, 11, 3, 9, 2, 2, 2, 19, 19, 3, 2, 2, 2, 2, 2, 2, 2, 5, 5, 3, 13, 13, 5, 2, 2, 7, 7, 3, 17, 17, 2, 2, 2, 31, 31, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 3, 23, 23, 2, 47, 47, 3, 2, 2, 2, 2, 2

Offset: 1

Rémy Sigrist, Aug 16 2025

nonn

			The first terms, alongside the first terms of A386482, are:
  n   a(n)  A386482(n)
  --  ----  ----------
   1     1           1
   2     2           2
   3     2           4
   4     3           6
   5     3           3
   6     3           9
   7     2          12
   8     2          10
   9     2           8
  10     7          14
  11     7           7
  12     3          21
  13     2          18
  14     4          16

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

Cf. A073734, A386482.

PARI
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a(n) = gcd(A386482(n), A386482(n+1)).

cp's OEIS Frontend

User: Rémy Sigrist

A387542 a(n) is the distance from the n-th term of A386482 to the nearest term of A386482 coprime to it.

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A387544 Even terms of A386482, halved, in order of occurrence.

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A387558 Numbers k such that A386482(k) is even.

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A387094 An infinite variant of the EKG sequence (A064413) such that for any n > 0, a(2*n) > a(2*n-1) and a(2*n) > a(2*n+1).

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A387059 Inverse permutation to A387058.

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A387058 Lexicographically earliest sequence of distinct nonnegative integers such that each term is a square number or belongs to a run of two consecutive terms summing to a square number.

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A387025 Start with the list of positive integers L_1 = (1, 2, ...); for n = 1, 2, ..., let m be the least integer > n such that L_n(n) divides L_n(m); L_{n+1}(k) = L_n(k) for any k <> m, L_{n+1}(m) = L_n(m)/L_n(n); a(n) = L_n(n).

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PARI

Formula

A387104 Split A386482 into maximal runs of consecutive decreasing terms; a(n) is the length of the n-th run.

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A387094 An infinite variant of the EKG sequence (A064413) such that for any n > 0, a(2n) > a(2n-1) and a(2n) > a(2n+1).