A268678 -id:A268678 - cp's OEIS frontend

A268677 Complement of A268678; numbers that do not occur in A268395.

2, 6, 9, 10, 12, 13, 14, 17, 21, 24, 25, 28, 29, 30, 33, 35, 38, 39, 43, 44, 45, 46, 49, 51, 54, 55, 58, 61, 62, 63, 67, 68, 71, 73, 76, 77, 78, 79, 80, 84, 85, 88, 91, 94, 95, 96, 97, 99, 100, 103, 107, 110, 111, 112, 114, 115, 118, 122, 125, 126, 128, 129, 130, 133, 137, 140, 141, 143, 144, 145, 146, 149, 152

Offset: 1

Antti Karttunen, Feb 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..32768

Cf. A268395, A268678.
Cf. A268712.
Cf. also A096346.

f[n_] := Which[n == 1, 0, OddQ@ #, 0, EvenQ@ #, 1 + f[#/2]] &@ Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]]; Complement[Range@ #, Union@ Accumulate@ Array[f, {#}]] &@ 152 (* Michael De Vlieger, Feb 12 2016, after Jan Mangaldan at A006068 *)

A268680 Least monotonic left inverse of A268678.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 4, 5, 6, 6, 6, 7, 7, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 14, 14, 15, 16, 16, 16, 16, 17, 18, 18, 19, 19, 20, 21, 21, 21, 22, 23, 24, 24, 24, 24, 24, 25, 26, 26, 27, 27, 28, 29, 29, 29, 30, 31, 31, 32, 33, 33, 33, 33, 34, 35, 36, 36, 36, 37, 38, 38, 39, 39, 40, 41, 41, 41, 41, 41, 41, 42, 43, 44

Offset: 0

Antti Karttunen, Feb 11 2016

nonn

a(n) = number of distinct nonzero values A268395 that occur in range [0 .. n].

Each n occurs A268679(n+1) times.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A268678, A268679, A268395.

Cf. also A268711.

Other identities. For all n >= 0:

a(A268678(n)) = n.

A268712 Permutation of natural numbers: a(1) = 1, a(2n) = A268677(a(n)), a(2n+1) = A268678(1+a(n)).

Original entry on oeis.org

1, 2, 3, 6, 4, 9, 5, 13, 11, 10, 7, 21, 18, 12, 8, 29, 23, 25, 20, 24, 19, 14, 15, 45, 40, 39, 34, 28, 22, 17, 16, 62, 56, 49, 42, 54, 48, 44, 37, 51, 47, 43, 36, 30, 26, 33, 27, 95, 87, 84, 75, 80, 74, 73, 65, 61, 53, 46, 41, 38, 32, 35, 31, 129, 120, 115, 108, 100, 93, 88, 82, 112, 105, 99, 92, 94, 86, 78, 70

Offset: 1

Antti Karttunen, Feb 11 2016

This sequence can be represented as a binary tree. Each left hand child is produced as A268677(n), and each right hand child as A268678(1+n), when the parent node contains n:
                                     |
                  ...................1...................
                 2                                       3
       6......../ \........4                   9......../ \........5
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  13       11         10       7          21       18         12       8
29 23    25  20     24  19   14 15      45  40   39  34     28  22   17 16
etc.

Antti Karttunen, Table of n, a(n) for n = 1..16383
Index entries for sequences that are permutations of the natural numbers

Inverse: A268711.

Cf. A268677, A268678.

a(1) = 1, after which: a(2n) = A268677(a(n)), a(2n+1) = A268678(1+a(n)).

A268711 Permutation of natural numbers: a(1) = 1, a(A268677(n)) = 2a(n), a(A268678(n+1)) = 1+(2a(n)).

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 11, 15, 6, 10, 9, 14, 8, 22, 23, 31, 30, 13, 21, 19, 12, 29, 17, 20, 18, 45, 47, 28, 16, 44, 63, 61, 46, 27, 62, 43, 39, 60, 26, 25, 59, 35, 42, 38, 24, 58, 41, 37, 34, 91, 40, 95, 57, 36, 90, 33, 89, 94, 127, 123, 56, 32, 88, 93, 55, 125, 126, 122, 87, 79, 92, 121, 54, 53, 51, 124, 86, 78, 120

Offset: 1

Antti Karttunen, Feb 11 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A268712.

Cf. A268677, A268678, A268680.

a(1) = 1, for n > 1: if A268680(n) = A268680(n-1) [when n is in A268677] a(n) = 2*a(n-A268680(n)), otherwise [when n is in A268678] a(n) = 1 + 2*a(A268680(n)-1).

A268395 Partial sums of A268389.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 4, 4, 5, 7, 7, 8, 8, 8, 11, 11, 15, 16, 16, 18, 18, 18, 19, 20, 20, 20, 22, 22, 23, 26, 26, 26, 27, 31, 31, 32, 32, 32, 34, 36, 36, 36, 37, 37, 40, 41, 41, 42, 42, 42, 47, 47, 48, 50, 50, 50, 52, 53, 53, 56, 56, 56, 57, 57, 59, 60, 60, 64, 64, 64, 65, 66, 66, 66, 69, 69, 70, 72, 72, 74, 74, 74, 75, 75, 81

Offset: 0

Antti Karttunen, Feb 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A048631, A268389, A268672.
Cf. A268678 (with duplicates removed), A268677 (numbers that do not occur here).
Cf. A268708, A268709, A268710.
Cf. also A054861.

f[n_] := Which[n == 1, 0, OddQ@ #, 0, EvenQ@ #, 1 + f[#/2]] &@ Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]]; Accumulate@ Array[f, {85}] (* Michael De Vlieger, Feb 12 2016, after Jan Mangaldan at A006068 *)

a(0) = 0, for n >= 1, a(n) = A268389(n) + a(n-1).
Other identities. For all n >= 0:
a(n) = A268389(A048631(n)).
a(n) = n - A268672(n).

A268679 a(n) = A268389(A001969(1+n)); A268389 without its zero terms.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 3, 4, 1, 2, 1, 1, 2, 1, 3, 1, 4, 1, 2, 2, 1, 3, 1, 1, 5, 1, 2, 2, 1, 3, 1, 2, 1, 4, 1, 1, 3, 1, 2, 2, 1, 6, 1, 1, 3, 1, 2, 1, 2, 1, 5, 3, 1, 2, 1, 1, 2, 1, 4, 3, 1, 2, 1, 1, 2, 1, 3, 4, 1, 2, 1, 1, 2, 1, 3, 5, 1, 2, 1, 2, 1, 3, 1, 1, 6, 1, 2, 2, 1, 3, 1, 1, 4, 1, 2, 1, 3, 1, 2, 2, 1, 5, 1, 1, 3, 1, 2, 2, 1, 4, 1, 3, 1, 2, 1, 1, 2, 1, 4, 3

Offset: 1

Antti Karttunen, Feb 10 2016

Antti Karttunen, Table of n, a(n) for n = 1..32768

Cf. A001969, A268389.

Cf. A268678 (partial sums), A268680.

f[n_] := Which[n == 1, 0, OddQ@ #, 0, EvenQ@ #, 1 + f[#/2]] &@ Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]];
Array[f, {240}] /. 0 -> Nothing (* Michael De Vlieger, Feb 12 2016, after Jan Mangaldan at A006068 *)

(define (A268679 n) (A268389 (A001969 (+ 1 n))))

a(n) = A268389(A001969(1+n)).

cp's OEIS Frontend

A268677 Complement of A268678; numbers that do not occur in A268395.

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A268680 Least monotonic left inverse of A268678.

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A268712 Permutation of natural numbers: a(1) = 1, a(2n) = A268677(a(n)), a(2n+1) = A268678(1+a(n)).

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A268711 Permutation of natural numbers: a(1) = 1, a(A268677(n)) = 2*a(n), a(A268678(n+1)) = 1+(2*a(n)).

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A268395 Partial sums of A268389.

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A268679 a(n) = A268389(A001969(1+n)); A268389 without its zero terms.

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A268711 Permutation of natural numbers: a(1) = 1, a(A268677(n)) = 2a(n), a(A268678(n+1)) = 1+(2a(n)).