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.

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A270199 Self-inverse permutation of natural numbers: a(1) = 1, a(A269389(1+n)) = A269399(a(n)), a(A269399(n)) = A269389(1+a(n)).

Original entry on oeis.org

1, 3, 2, 9, 6, 5, 30, 15, 4, 16, 12, 11, 93, 45, 8, 10, 46, 48, 34, 33, 266, 124, 26, 31, 127, 23, 154, 99, 97, 7, 24, 727, 20, 19, 352, 80, 94, 357, 68, 141, 69, 446, 278, 272, 14, 17, 70, 18, 71, 73, 1902, 54, 61, 52, 946, 232, 267, 957, 197, 408, 53, 199, 1174, 763, 407, 751, 186, 39, 41, 47, 49, 202, 50, 204, 210, 4724, 164, 192, 182, 36
Offset: 1

Views

Author

Antti Karttunen, Mar 16 2016

Keywords

Links

Crossrefs

Cf. A137264, A269362, A269389, A269399.
Related or similar permutations: A267107, A270193, A270194.

Formula

a(1) = 1, for n > 1, if A137264(6+n) = 0 [when n is in A269399], a(n) = A269389(1+a(n-A269362(n))), otherwise a(n) = A269399(a(A269362(n)-1)).

A270193 Permutation of natural numbers: a(1) = 1, a(A269389(1+n)) = 2 * a(n), a(A269399(n)) = 1 + 2*a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 7, 9, 12, 11, 16, 20, 13, 14, 17, 18, 24, 22, 32, 40, 26, 28, 34, 21, 36, 48, 44, 15, 19, 64, 25, 23, 80, 52, 56, 68, 42, 33, 41, 72, 96, 88, 27, 30, 38, 29, 35, 37, 128, 49, 50, 46, 160, 104, 112, 136, 84, 66, 45, 82, 144, 192, 65, 176, 81, 53, 54, 57, 60, 76, 58, 70, 74, 256, 98, 69, 100, 43, 92
Offset: 1

Views

Author

Antti Karttunen, Mar 16 2016

Keywords

Links

Crossrefs

Inverse: A270194.
Cf. A137264, A269362, A269389, A269399.
Similar permutations: A270199, A270201 (compare the scatter plots).

Formula

a(1) = 1, for n > 1, if A137264(6+n) = 0 [when n is in A269399], a(n) = 1 + 2*a(n-A269362(n)), otherwise a(n) = 2 * a(A269362(n)-1).

A270194 Permutation of natural numbers: a(1) = 1, a(2n) = A269389(1+a(n)), a(2n+1) = A269399(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 7, 10, 8, 12, 11, 15, 16, 30, 13, 17, 18, 31, 14, 26, 20, 34, 19, 33, 23, 45, 24, 48, 46, 93, 21, 40, 25, 49, 27, 50, 47, 94, 22, 41, 39, 80, 29, 61, 54, 99, 28, 52, 53, 97, 36, 68, 69, 124, 37, 70, 73, 154, 71, 127, 141, 266, 32, 65, 60, 112, 38, 78, 74, 155, 42, 85, 75, 158, 72, 133, 142, 267, 35, 67, 62, 113, 59
Offset: 1

Views

Author

Antti Karttunen, Mar 16 2016

Keywords

Comments

This sequence can be represented as a binary tree. Each left hand child is produced as A269389(1+n), and each right hand child as A269399(n), when the parent node contains n:
1
................../ \..................
2 3
4......../ \........5 6......../ \........9
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
7 10 8 12 11 15 16 30
13 17 18 31 14 26 20 34 19 33 23 45 24 48 46 93
etc.

Links

Crossrefs

Inverse: A270193.
Cf. A269389, A269399.
Similar permutations: A270199, A270202.

Formula

a(1) = 1, a(2n) = A269389(1+a(n)), a(2n+1) = A269399(a(n)).

A270190 Numbers n for which prime(n+1)-prime(n) is a multiple of three.

Original entry on oeis.org

9, 11, 15, 16, 18, 21, 23, 32, 36, 37, 39, 40, 46, 47, 51, 54, 55, 56, 58, 67, 71, 73, 74, 76, 84, 86, 91, 96, 97, 99, 100, 102, 103, 105, 107, 108, 110, 111, 114, 118, 119, 121, 123, 129, 130, 133, 139, 160, 161, 164, 165, 167, 168, 170, 174, 179, 180, 184, 185, 187, 188, 194, 195, 197, 199, 200, 202, 203, 205, 208, 210
Offset: 1

Views

Author

Antti Karttunen, Mar 16 2016

Keywords

Comments

Numbers n for which A001223(n) = 0 modulo 3.
See comments in A270189 and A269364.
Equivalently, numbers n for which prime(n+1)-prime(n) is a multiple of six. See A276414 for runs of increasing length of consecutive integers. - M. F. Hasler, Sep 03 2016

Examples

			9 is present as the difference between A000040(9+1) = 29 and A000040(9) = 23 is 6, a multiple of three.

Links

Crossrefs

Complement: A270189.
Positions of zeros in A137264.
Left inverse: A269850.
Cf. also A001223, A269364, A270191, A270192, A269399.

Programs

  • Mathematica
    Select[Range@ 210, Divisible[Prime[# + 1] - Prime@ #, 3] &] (* Michael De Vlieger, Mar 17 2016 *)
PrimePi/@Select[Partition[Prime[Range[350]],2,1],Divisible[#[[2]]-#[[1]], 3]&][[All,1]] (* Harvey P. Dale, Jul 11 2017 *)
  • PARI
    isok(n) = ((prime(n+1) - prime(n)) % 3) == 0; \\ Michel Marcus, Mar 17 2016

Formula

Other identities. For all n >= 1:
a(n) = A269399(n) + 6.
A269850(a(n)) = n.

A269389 Numbers n for which prime(n+7)-prime(n+6) is not a multiple of three.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 11, 13, 14, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 32, 35, 36, 37, 38, 39, 42, 43, 44, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 69, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 86, 87, 88, 89, 92, 95, 98, 100, 103, 106, 107, 109, 110, 111, 114, 116, 118, 119, 120, 121, 122, 125
Offset: 1

Views

Author

Antti Karttunen, Mar 16 2016

Keywords

Links

Crossrefs

Complement: A269399.
Left inverse: A269362.
Cf. A270189.
Cf. also A270199.

Programs

Formula

a(n) = A270189(6+n) - 6.
Other identities. For all n >= 1:
A269362(a(n)) = n.
Showing 1-5 of 5 results.

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