cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 44 results. Next

A249777 Let z = A084937: a(n) = number of numbers <= z(n) that are != z(k) for k=1..n-1 and not coprime to z(n-1) and z(n-2).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 2, 3, 0, 5, 6, 0, 6, 7, 3, 2, 10, 1, 5, 6, 3, 7, 10, 0, 10, 13, 0, 13, 16, 0, 5, 15, 1, 11, 16, 1, 6, 15, 2, 11, 16, 4, 16, 17, 2, 17, 12, 5, 16, 17, 3, 17, 18, 0, 20, 21, 0, 23, 28, 0, 23, 27, 1, 15, 30, 3, 17, 26, 4, 19, 28, 2, 21
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 05 2014

Keywords

Comments

a(A249684(n)) = 0.

Examples

			.   n | A084937(n) |    unused numbers less than A084937(n)   | a(n)
. ----+------------+------------------------------------------+-----
.   3 |          3 |  _                                       |    0
.   4 |          5 |  4                                       |    1
.   5 |          4 |  _                                       |    0
.   6 |          7 |  6                                       |    1
.   7 |          9 |  6, 8                                    |    2
.   8 |          8 |  6                                       |    1
.   9 |         11 |  6, 10                                   |    2
.  10 |         13 |  6, 10, 12                               |    3
.  11 |          6 |  _                                       |    0
.  12 |         17 |  10, 12, 14, 15, 16                      |    5
.  13 |         19 |  10, 12, 14, 15, 16, 18                  |    6
.  14 |         10 |  _                                       |    0
.  15 |         21 |  12, 14, 15, 16, 18, 20                  |    6
.  16 |         23 |  12, 14, 15, 16, 18, 20, 22              |    7
.  17 |         16 |  12, 14, 15                              |    3
.  18 |         15 |  12, 14                                  |    2
.  19 |         29 |  12, 14, 18, 20, 22, 24, 25, 26, 27, 28  |   10
.  20 |         14 |  12                                      |    1
.  21 |         25 |  12, 18, 20, 22, 24                      |    5
.  22 |         27 |  12, 18, 20, 22, 24, 26                  |    6
.  23 |         22 |  12, 18, 20                              |    3
.  24 |         31 |  12, 18, 20, 24, 26, 28, 30              |    7
.  25 |         35 |  12, 18, 20, 24, 26, 28, 30, 32, 33, 34  |   10 .

Links

Crossrefs

Cf. A084937, A249684.

Programs

  • Haskell
    a249777 n = a249777_list !! (n-1)
a249777_list = 0 : 0 : f 2 1 [3..] where
   f x y zs = g zs 0 where
      g (u:us) w | gcd y u > 1 || gcd x u > 1 = g us (w + 1)
                 | otherwise = w : f u x (delete u zs)

A249856 Let z = A084937: a(n) = number of odd numbers <= z(n) that are != z(k) for k=1..n-1 and not coprime to z(n-1) and z(n-2).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 2, 0, 2, 3, 0, 0, 2, 0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 4, 0, 2, 3, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 09 2014

Keywords

Comments

a(n) = A249777(n) - A249857(n).

Crossrefs

Cf. A084937, A249777, A249857, A249858.
For a different way to look at the missing numbers in A084937, see A249686, A250099, A250100.

Programs

  • Haskell
    a249856 = sum . map (flip mod 2) . (uss !!)
uss = [] : [] : [] : f 2 1 [3..] where
   f x y zs = g zs [] where
      g (v:vs) ws | gcd v y > 1 || gcd v x > 1 = g vs (v : ws)
                  | otherwise = ws : f v x (delete v zs)

A249857 Let z = A084937: a(n) = number of even numbers <= z(n) that are != z(k) for k=1..n-1 and not coprime to z(n-1) and z(n-2).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 2, 3, 0, 4, 5, 0, 5, 6, 2, 2, 8, 1, 5, 6, 3, 7, 9, 0, 9, 11, 0, 11, 13, 0, 5, 13, 1, 10, 14, 1, 6, 14, 2, 11, 15, 4, 15, 16, 2, 16, 12, 5, 16, 17, 3, 17, 18, 0, 19, 20, 0, 21, 24, 0, 21, 24, 1, 15, 26, 3, 17, 24, 4, 19, 26, 2, 21, 26
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 09 2014

Keywords

Comments

a(n) = A249777(n) - A249856(n).

Examples

			.     |            |     unused even numbers <  A084937(n)    |
.   n | A084937(n) |     [uncounted odd terms in brackets]    | a(n)
. ----+------------+------------------------------------------+-----
.   3 |          3 |  _                                       |    0
.   4 |          5 |  4                                       |    1
.   5 |          4 |  _                                       |    0
.   6 |          7 |  6                                       |    1
.   7 |          9 |  6,8                                     |    2
.   8 |          8 |  6                                       |    1
.   9 |         11 |  6,10                                    |    2
.  10 |         13 |  6,10,12                                 |    3
.  11 |          6 |  _                                       |    0
.  12 |         17 |  10,12,14,[15],16                        |    4
.  13 |         19 |  10,12,14,[15],16,18                     |    5
.  14 |         10 |  _                                       |    0
.  15 |         21 |  12,14,[15],16,18,20                     |    5
.  16 |         23 |  12,14,[15],16,18,20,22                  |    6
.  17 |         16 |  12,14,[15]                              |    2
.  18 |         15 |  12,14                                   |    2
.  19 |         29 |  12,14,18,20,22,24,[25],26,[27],28       |    8
.  20 |         14 |  12                                      |    1
.  21 |         25 |  12,18,20,22,24                          |    5
.  22 |         27 |  12,18,20,22,24,26                       |    6
.  23 |         22 |  12,18,20                                |    3
.  24 |         31 |  12,18,20,24,26,28,30                    |    7
.  25 |         35 |  12,18,20,24,26,28,30,32,[33],34         |    9 .

Crossrefs

Cf. A084937, A249777, A249856, A249858.
For a different way to look at the missing numbers in A084937, see A249686, A250099, A250100.

Programs

  • Haskell
    a249857 = sum . map ((1 -) . flip mod 2) . (uss !!)
-- See A249856 for definition of uss.

A249686 After A084937(n) has been computed, let m = largest term so far in A084937. Then a(n) = number of positive integers < m that are missing from A084937 at this point.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 2, 3, 2, 5, 6, 5, 6, 7, 6, 5, 10, 9, 8, 7, 6, 7, 10, 9, 10, 13, 12, 13, 16, 15, 14, 15, 14, 13, 16, 15, 14, 15, 14, 13, 16, 15, 16, 17, 16, 17, 16, 15, 16, 17, 16, 17, 18, 17, 20, 21, 20, 23, 28, 27, 26, 27, 26, 25, 30, 29, 28, 27, 26, 25, 28
Offset: 1

Views

Author

N. J. A. Sloane, Nov 05 2014

Keywords

Comments

Running count of missing numbers in A084937.
It appears that at any point, the number of missing even numbers from A084937 is always much larger than the number of missing odd numbers. It would be nice to have a more precise statement of this property.
In this regard, it would be helpful to have two further sequences, one giving the number of even missing numbers at each point, the other giving the number of odd missing numbers. These are now A250099, A250100. See also A249777, A249856, A249867.

Examples

			After step 7 of A084937, here is what we have:
1 2 3 4 5 6 7 ... n
1 2 3 5 4 7 9 ... A084937(n)
so m = 9, and the missing numbers < 9 are 6 and 8, so a(7) = 2.

Links

Crossrefs

Cf. A084937, A250099, A250100. See A249777, A249856, A249857, A249858 for another way of looking at this question.

A250099 After A084937(n) has been computed, let m = largest term so far in A084937. Then a(n) = number of positive even integers < m that are missing from A084937 at this point.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 2, 3, 2, 4, 5, 4, 5, 6, 5, 5, 8, 7, 7, 7, 6, 7, 9, 8, 9, 11, 10, 11, 13, 12, 12, 13, 12, 12, 14, 13, 13, 14, 13, 13, 15, 14, 15, 16, 15, 16, 16, 15, 16, 17, 16, 17, 18, 17, 19, 20, 19, 21, 24, 23, 23, 24, 23, 23, 26, 25, 25, 25, 24, 24, 26, 25
Offset: 1

Views

Author

N. J. A. Sloane, Nov 12 2014

Keywords

Comments

Running count of missing even numbers in A084937.

Examples

			After step 7 of A084937, here is what we have:
1 2 3 4 5 6 7 ... n
1 2 3 5 4 7 9 ... A084937(n)
so m = 9, and the missing even numbers < 9 are 6 and 8, so a(7) = 2.

Links

Crossrefs

Cf. A084937, A249686, A250100. See A249777, A249856, A249857 for another way of looking at this question.

A250100 After A084937(n) has been computed, let m = largest term so far in A084937. Then a(n) = number of positive odd integers < m that are missing from A084937 at this point.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 2, 2, 1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 1, 2, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 4, 4, 3, 3, 3, 2, 4, 4, 3, 2, 2, 1, 2, 2, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

N. J. A. Sloane, Nov 12 2014

Keywords

Comments

Running count of missing odd numbers in A084937.

Examples

			After step 7 of A084937, here is what we have:
1 2 3 4 5 6 7 ... n
1 2 3 5 4 7 9 ... A084937(n)
so m = 9, and the missing numbers < 9 are 6 and 8, both even, so a(7) = 0.

Links

Crossrefs

Cf. A084937, A249686, A250099. See A249777, A249856, A249857 for another way of looking at this question.

A249680 A084937(3n+1).

Original entry on oeis.org

1, 5, 9, 13, 19, 23, 29, 27, 35, 41, 47, 49, 53, 55, 59, 63, 57, 69, 73, 79, 89, 91, 97, 95, 101, 103, 107, 109, 113, 117, 121, 127, 137, 139, 149, 151, 145, 155, 157, 161, 163, 167, 169, 173, 179, 181, 191, 193, 199, 209, 211, 221, 215, 223, 227, 229, 233, 235
Offset: 0

Views

Author

N. J. A. Sloane, Nov 03 2014

Keywords

Links

Crossrefs

Cf. A084937, A249681-A249683.

Programs

A249683 a(n) = A084937(3n+2)/2.

Original entry on oeis.org

1, 2, 4, 3, 5, 8, 7, 11, 6, 9, 10, 13, 14, 16, 19, 17, 22, 20, 12, 15, 18, 23, 26, 28, 25, 29, 31, 32, 34, 37, 24, 21, 27, 30, 35, 38, 41, 43, 40, 44, 46, 47, 49, 50, 52, 33, 36, 39, 42, 48, 45, 53, 56, 55, 58, 59, 61, 62, 64, 67, 68, 71, 65, 70, 73, 76, 51
Offset: 0

Views

Author

N. J. A. Sloane, Nov 03 2014

Keywords

Links

Crossrefs

Cf. A084937, A249680-A249682.

Programs

A249684 Numbers that take a record number of steps before they appear in A084937.

Original entry on oeis.org

1, 2, 3, 5, 11, 14, 26, 29, 32, 56, 59, 62, 95, 98, 101, 137, 140, 143, 146, 152, 200, 203, 206, 209, 215, 221, 224, 281, 287, 290, 293, 296, 299, 302, 305, 365, 371, 374, 377, 380, 383, 386, 392, 398, 401, 485, 488, 491, 497, 500, 503, 506, 509, 512, 518, 521, 533, 614, 620, 623, 626, 629, 632, 635
Offset: 1

Views

Author

N. J. A. Sloane, Nov 05 2014

Keywords

Comments

Records in A084933.
A249777(a(n)) = 0. - Reinhard Zumkeller, Nov 05 2014

Links

Crossrefs

Cf. A084933, A084937, A249685, A249777.

Programs

  • Haskell
    a249684 n = a249684_list !! (n-1)
a249684_list = filter ((== 0) . a249777) [1..]
-- Reinhard Zumkeller, Nov 05 2014

A084933 Inverse of A084937.

Original entry on oeis.org

1, 2, 3, 5, 4, 11, 6, 8, 7, 14, 9, 26, 10, 20, 18, 17, 12, 29, 13, 32, 15, 23, 16, 56, 21, 35, 22, 38, 19, 59, 24, 41, 33, 47, 25, 62, 27, 44, 39, 53, 28, 95, 30, 50, 36, 65, 31, 92, 34, 74, 42, 68, 37, 98, 40, 71, 49, 77, 43, 101, 45, 80, 46, 83, 48, 137, 51, 86, 52
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 13 2003

Keywords

Comments

Empirically, A084937 is a permutation of the natural numbers. Assuming that is true, we will have a(A084937(n)) = A084937(a(n)) = n. (Comment revised by N. J. A. Sloane, Nov 05 2014)

Links

Crossrefs

Cf. A084937. For records see A249684, A249685.

Programs

  • Haskell
    import Data.List (elemIndex)
import Data.Maybe (fromJust)
a084933 n = (fromJust $ elemIndex n a084937_list) + 1
-- Reinhard Zumkeller, Jan 28 2012
  • Mathematica
    f[s_] := Block[{k = 1, l = Take[s, -2]}, While[ Union[ GCD[k, l]] != {1} || MemberQ[s, k], k++]; Append[s, k]]; Ordering@ Nest[f, {1, 2}, 100] (* Robert G. Wilson v, Jun 26 2011 *)
Showing 1-10 of 44 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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