A250469 -id:A250469 - cp's OEIS frontend

A252755 Tree of Eratosthenes, mirrored: a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A250469(a(n)).

1, 2, 4, 3, 8, 9, 6, 5, 16, 21, 18, 25, 12, 15, 10, 7, 32, 45, 42, 55, 36, 51, 50, 49, 24, 33, 30, 35, 20, 27, 14, 11, 64, 93, 90, 115, 84, 123, 110, 91, 72, 105, 102, 125, 100, 147, 98, 121, 48, 69, 66, 85, 60, 87, 70, 77, 40, 57, 54, 65, 28, 39, 22, 13, 128, 189, 186, 235, 180, 267, 230, 203, 168, 249, 246, 305, 220, 327, 182, 187, 144

Offset: 0

Antti Karttunen, Jan 02 2015

This sequence can be represented as a binary tree. Each child to the left is obtained by doubling the parent, and each child to the right is obtained by applying A250469 to the parent:
                                     1
                                     |
                  ...................2...................
                 4                                       3
       8......../ \........9                   6......../ \........5
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  16       21         18       25         12       15         10       7
32  45   42  55     36  51   50  49     24  33   30  35     20  27   14 11
etc.
Sequence A252753 is the mirror image of the same tree. A253555(n) gives the distance of n from 1 in both trees.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences that are permutations of the natural numbers

Inverse: A252756.
Row sums: A253787, products: A253788.
Similar permutations: A163511, A252753, A054429, A163511, A250245, A269865.
Cf. also: A249814 (Compare the scatterplots).
Cf. A083221, A250469, A253555.

(* b = A250469 *) b[1] = 1; b[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[1, 1]]; For[k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1 + 2 == k2, Return[m2]]]];
a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ[n], 2 a[n/2], b[a[(n-1)/2]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2016 *)

a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A250469(a(n)).
As a composition of related permutations:
a(n) = A252753(A054429(n)).
a(n) = A250245(A163511(n)).

A252753 Tree of Eratosthenes: a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 21, 16, 11, 14, 27, 20, 35, 30, 33, 24, 49, 50, 51, 36, 55, 42, 45, 32, 13, 22, 39, 28, 65, 54, 57, 40, 77, 70, 87, 60, 85, 66, 69, 48, 121, 98, 147, 100, 125, 102, 105, 72, 91, 110, 123, 84, 115, 90, 93, 64, 17, 26, 63, 44, 95, 78, 81, 56, 119, 130, 159, 108, 145, 114, 117, 80

Offset: 0

Antti Karttunen, Jan 02 2015

This sequence can be represented as a binary tree. Each child to the left is obtained by applying A250469 to the parent, and each child to the right is obtained by doubling the parent:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........6                   9......../ \........8
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  7       10         15       12         25       18         21       16
11 14   27  20     35  30   33  24     49  50   51  36     55  42   45  32
etc.
Sequence A252755 is the mirror image of the same tree. A253555(n) gives the distance of n from 1 in both trees.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A252754.
Row sums: A253787, products: A253788.
Fixed points of a(n-1): A253789.
Similar permutations: A005940, A252755, A054429, A250245.
Cf. also A001248, A001511, A055396, A083221, A181565, A250469, A253555.

(* b = A250469 *)
b[1] = 1; b[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[ 1, 1]]; For[k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1+2 == k2, Return[m2]]]];
a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ[n], b[a[n/2]], 2 a[(n-1)/2]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2016 *)

a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).
As a composition of related permutations:
a(n) = A252755(A054429(n)).
a(n) = A250245(A005940(1+n)).
Other identities. For all n >= 1:
A055396(a(n)) = A001511(n). [A005940 has the same property.]
a(A003945(n)) = A001248(n) for n>=1. - Peter Luschny, Jan 13 2015

A266403 Self-inverse permutation of natural numbers: a(n) = A250470(A263273(A250469(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 8, 17, 6, 13, 10, 11, 20, 9, 14, 71, 22, 7, 26, 19, 12, 23, 16, 21, 24, 41, 18, 53, 28, 31, 56, 29, 38, 107, 58, 67, 74, 61, 32, 197, 40, 25, 68, 59, 50, 137, 64, 73, 62, 49, 44, 227, 76, 27, 80, 55, 30, 89, 34, 43, 66, 37, 48, 91, 46, 69, 60, 35, 42, 65, 70, 15, 78, 47, 36, 119, 52

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..6560
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A250469, A250470, A263273.
Cf. A265369, A265904, A266190, A266401 (other conjugates or similar derivations of A263273).
Cf. also A250471, A250472, A264996, A266404, A266415, A266416, A266645, A266646.

(define (A266403 n) (A250470 (A263273 (A250469 n))))

a(n) = A250470(A263273(A250469(n))).
As a composition of related permutations:
a(n) = A266415(A266645(n)) = A266646(A266416(n)).
a(n) = A250472(A264996(A250471(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A266645 Permutation of natural numbers: a(n) = A064989(A250469(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 14, 13, 22, 15, 12, 17, 26, 19, 34, 21, 20, 23, 38, 25, 18, 33, 16, 29, 46, 31, 58, 39, 28, 35, 30, 37, 62, 51, 44, 41, 74, 43, 82, 57, 24, 47, 86, 49, 50, 27, 52, 53, 94, 55, 42, 69, 68, 59, 106, 61, 118, 87, 40, 65, 66, 67, 122, 45, 76, 71, 134, 73, 142, 93, 36, 77, 70, 79, 146, 111, 32, 83, 158, 85, 78, 123

Offset: 1

Antti Karttunen, Jan 02 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A266646.
Cf. A000035, A003961, A020639, A055396, A064989, A250469.
Related permutations: A266403, A266416, A249817, A249818.

f[n_] := Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[{m, n}, f[Lookup[s, g[n] + 1][[m]] - Boole[n == 1]]][#1, First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 09 2017, Version 10 *)

a(n) = A064989(A250469(n)).
a(n) = A064989(A249817(A003961(A249818(n)))).
As a composition of related permutations:
a(n) = A266416(A266403(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A020639(a(n)) = A020639(n). [More generally, it preserves the smallest prime dividing n.]
A055396(a(n)) = A055396(n).

A269172 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A269380(2n+1))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 20, 21, 22, 19, 24, 23, 26, 27, 28, 29, 30, 49, 32, 33, 34, 35, 36, 31, 50, 39, 40, 37, 42, 41, 44, 45, 38, 43, 48, 55, 46, 51, 52, 47, 54, 121, 56, 57, 58, 77, 60, 53, 98, 63, 64, 65, 66, 59, 68, 69, 70, 61, 72, 169, 62, 75, 100, 67, 78, 85, 80, 81

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2244
Index entries for sequences that are permutations of the natural numbers

Inverse: A269171.
Cf. A000035, A003309, A008578, A083221, A250469, A260738, A260739, A269380.
Related or similar permutations: A260741, A260742, A269356, A269358, A255422.
Cf. also A269394 (a(3n)/3) and A269396.
Differs from A255408 for the first time at n=38, where a(38) = 50, while A255408(38) = 38.

a(1) = 1, then after for even n, a(n) = 2*a(n/2), and for odd n, A250469(a(A269380(n))).
a(1) = 1, for n > 1, a(n) = A083221(A260738(n), a(A260739(n))).
As a composition of other permutations:
a(n) = A252755(A269386(n)).
a(n) = A252753(A269388(n)).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
a(A003309(n)) = A008578(n). [Maps Ludic numbers to noncomposites.]

A269865 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(1+a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 11, 18, 27, 16, 25, 14, 21, 20, 13, 30, 45, 24, 17, 22, 33, 36, 23, 54, 81, 32, 19, 50, 75, 28, 35, 42, 63, 40, 55, 26, 39, 60, 37, 90, 135, 48, 49, 34, 51, 44, 29, 66, 99, 72, 41, 46, 69, 108, 91, 162, 243, 64, 85, 38, 57, 100, 125, 150, 225, 56, 31, 70, 105, 84, 47, 126, 189, 80, 43, 110, 165, 52

Offset: 1

Antti Karttunen, Mar 12 2016

This sequence can be represented as a binary tree. When the parent contains n, the left hand child contains 2n, while the value of right hand child is obtained by applying A250469(1+n):
                                    1
                                    |
                 ................../ \..................
                2                                       3
      4......../ \........5                   6......../ \........9
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  8       7          10       15         12       11         18       27
16 25   14 21      20  13   30  45     24  17   22  33     36  23   54  81
etc.
Note how all nodes with odd n have a right hand child with value 3n.

Antti Karttunen, Table of n, a(n) for n = 1..6142
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences that are permutations of the natural numbers

Inverse: A269866.
Cf. A250469.
Related or similar permutations: A269359, A269863, A269864, A269867, A246375, A249814, A252755, A270195.

a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(1+a(n)).

A280692 a(n) = A003961(n) - A250469(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 0, -6, 0, 12, 0, -6, 0, 36, 0, 24, 0, 6, 0, -24, 0, 66, 0, -24, 60, 18, 0, 18, 0, 150, -20, -42, 0, 120, 0, -42, -10, 72, 0, 42, 0, -12, 60, -48, 0, 264, 0, 0, -30, 0, 0, 216, 0, 132, -30, -78, 0, 138, 0, -72, 120, 540, 0, 0, 0, -30, -30, 24, 0, 462, 0, -96, 60, -18, 0, 24, 0, 330, 420, -114, 0, 246

Offset: 1

Antti Karttunen, Mar 08 2017

sign

Antti Karttunen, Table of n, a(n) for n = 1..10067

Cf. A003961, A250469, A249820, A280492, A280497, A280498, A280702.
Cf. A280693 (gives the positions of zeros).
Cf. also arrays A083221 and A246278.

f[n_] := f[n] = Which[n == 1, 1, PrimeQ@ n, NextPrime@ n, True, Times @@ Replace[FactorInteger[n], {p_, e_} :> f[p]^e, 1]]; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[{m, n}, f@ n - Lookup[s, g[n] + 1][[m]] + Boole[n == 1]][#1, First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 09 2017, Version 10 *)

(define (A280692 n) (- (A003961 n) (A250469 n)))

a(n) = A003961(n) - A250469(n).

A250471 Permutation of natural numbers: a(n) = (A250469(n) + 1) / 2.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 6, 11, 13, 14, 7, 17, 9, 20, 18, 23, 10, 26, 12, 29, 28, 32, 15, 35, 25, 38, 33, 41, 16, 44, 19, 47, 43, 50, 39, 53, 21, 56, 48, 59, 22, 62, 24, 65, 58, 68, 27, 71, 61, 74, 63, 77, 30, 80, 46, 83, 73, 86, 31, 89, 34, 92, 78, 95, 60, 98, 36, 101, 88, 104, 37, 107, 40, 110, 93, 113, 72, 116, 42, 119, 103, 122, 45, 125, 67, 128, 108, 131, 49

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..5002
Index entries for sequences that are permutations of the natural numbers

Inverse: A250472.

Cf. A048673, A250469.

(define (A250471 n) (/ (+ 1 (A250469 n)) 2))

a(n) = (A250469(n) + 1) / 2.

A253886 Permutation of even numbers: a(n) = A250469(n+1) - 1.

Original entry on oeis.org

0, 2, 4, 8, 6, 14, 10, 20, 24, 26, 12, 32, 16, 38, 34, 44, 18, 50, 22, 56, 54, 62, 28, 68, 48, 74, 64, 80, 30, 86, 36, 92, 84, 98, 76, 104, 40, 110, 94, 116, 42, 122, 46, 128, 114, 134, 52, 140, 120, 146, 124, 152, 58, 158, 90, 164, 144, 170, 60, 176, 66, 182, 154, 188, 118, 194, 70, 200, 174, 206, 72, 212, 78, 218, 184, 224, 142, 230, 82

Offset: 0

Antti Karttunen, Jan 18 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..5000

Cf. A250469, A253885, A249814.

(define (A253886 n) (- (A250469 (+ n 1)) 1))

a(n) = A250469(n+1) - 1.

A266416 Permutation of natural numbers: a(n) = A064989(A263273(A250469(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 17, 6, 13, 8, 11, 34, 9, 22, 71, 20, 7, 18, 19, 14, 23, 12, 21, 38, 41, 26, 53, 16, 31, 42, 29, 62, 107, 68, 67, 142, 61, 58, 197, 44, 25, 122, 59, 50, 137, 40, 73, 118, 49, 82, 227, 36, 33, 146, 55, 46, 89, 28, 43, 66, 37, 86, 91, 24, 45, 106, 35, 74, 65, 76, 15, 70, 47, 30, 119, 52

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..6562
Index entries for sequences that are permutations of the natural numbers

Inverse: A266415.
Cf. A000035, A064989, A250469, A263273.
Related permutations: A064216, A250471, A264985, A264996, A266403, A266645.

(define (A266416 n) (A064989 (A263273 (A250469 n))))

a(n) = A064989(A263273(A250469(n))).
As a composition of related permutations:
a(n) = A266645(A266403(n)).
a(n) = A064216(A264996(A250471(n))) = A064216(1+A264985(-1+A250471(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

cp's OEIS Frontend

A252755 Tree of Eratosthenes, mirrored: a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A250469(a(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A252753 Tree of Eratosthenes: a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A266403 Self-inverse permutation of natural numbers: a(n) = A250470(A263273(A250469(n))).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A266645 Permutation of natural numbers: a(n) = A064989(A250469(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A269172 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A269380(2n+1))).

Views

Author

Keywords

Links

Crossrefs

Formula

A269865 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(1+a(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A280692 a(n) = A003961(n) - A250469(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A250471 Permutation of natural numbers: a(n) = (A250469(n) + 1) / 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A253886 Permutation of even numbers: a(n) = A250469(n+1) - 1.

Views

Author

Keywords

Links

Crossrefs

Programs