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 1-6 of 6 results.

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)).

A269362 Least monotonic left inverse of A269389.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Mar 16 2016

Keywords

Comments

a(n) = number of terms of A269389 <= n.

Links

Crossrefs

Cf. A269389, A269849.
Cf. also A270193, A270199.

Formula

a(n) = A269849(6+n) - 6.
Other identities. For all n >= 1:
a(A269389(n)) = n.

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)).

A270189 Numbers n for which (prime(n+1)-prime(n)) is not a multiple of three.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 17, 19, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 38, 41, 42, 43, 44, 45, 48, 49, 50, 52, 53, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 75, 77, 78, 79, 80, 81, 82, 83, 85, 87, 88, 89, 90, 92, 93, 94, 95, 98, 101, 104, 106, 109, 112, 113, 115, 116, 117, 120, 122, 124, 125
Offset: 1

Views

Author

Antti Karttunen, Mar 16 2016

Keywords

Comments

See A269364 for the effect of the bias that favors these terms over the terms of A270190.

Links

Crossrefs

Complement: A270190.
Disjoint union of A270191 and A270192.
Positions of 1's and 2's in A137264.
Left inverse: A269849.
Cf. A000040, A001223, A269364, A269389.

Programs

  • Mathematica
    Select[Range@ 125, Mod[Prime[# + 1] - Prime@ #, 3] != 0 &] (* Michael De Vlieger, Mar 17 2016 *)
  • PARI
    isok(n) = ((prime(n+1) - prime(n)) % 3) != 0; \\ Michel Marcus, Mar 17 2016

Formula

Other identities. For all n >= 1:
A269849(a(n)) = n.

A269399 Numbers n for which prime(n+7)-prime(n+6) is a multiple of three.

Original entry on oeis.org

3, 5, 9, 10, 12, 15, 17, 26, 30, 31, 33, 34, 40, 41, 45, 48, 49, 50, 52, 61, 65, 67, 68, 70, 78, 80, 85, 90, 91, 93, 94, 96, 97, 99, 101, 102, 104, 105, 108, 112, 113, 115, 117, 123, 124, 127, 133, 154, 155, 158, 159, 161, 162, 164, 168, 173, 174, 178, 179, 181, 182, 188, 189, 191, 193, 194, 196, 197, 199, 202, 204, 208, 210
Offset: 1

Views

Author

Antti Karttunen, Mar 16 2016

Keywords

Examples

			3 is present as the difference between A000040(3+7) = 29 and A000040(3+6) = 23 is 6, a multiple of three.

Links

Crossrefs

Complement: A269389.
Cf. A137264, A270190.
Cf. also A270199.

Programs

  • Mathematica
    Select[Range@ 210, Divisible[Prime[# + 7] - Prime[# + 6], 3] &] (* Michael De Vlieger, Mar 18 2016 *)
  • PARI
    isok(n) = ((prime(n+7) - prime(n+6)) % 3) == 0; \\ Michel Marcus, Mar 18 2016

Formula

a(n) = A270190(n) - 6.
Showing 1-6 of 6 results.

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