0,2

This is a permutation of the positive integers.

From Antti Karttunen, Jun 20 2014: (Start)

Note the indexing: the domain starts from 0, while the range excludes zero, thus this is neither a bijection on the set of nonnegative integers nor on the set of positive natural numbers, but a bijection from the former set to the latter.

Apart from that discrepancy, this could be viewed as yet another "entanglement permutation" where the two complementary pairs to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with the complementary pair even numbers (taken straight) and odd numbers in the order they appear in A003961: (A005843/A003961). See also A246375 which has almost the same recurrence.

Note how the even bisection halved gives the same sequence back. (For a(0)=1, take ceiling of 1/2).

(End)

From Antti Karttunen, Dec 30 2017: (Start)

This irregular table 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 A003961 to the parent:

1

|

...................2...................

4 3

8......../ \........9 6......../ \........5

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

16 27 18 25 12 15 10 7

32 81 54 125 36 75 50 49 24 45 30 35 20 21 14 11

etc.

Sequence A005940 is obtained by scanning the same tree level by level in mirror image fashion. Also in binary trees A253563 and A253565 the terms on level of the tree are some permutation of the terms present on the level n of this tree. A252464(n) gives the distance of n from 1 in all these trees, and A252463 gives the parent of the node containing n.

A252737(n) gives the sum and A252738(n) the product of terms on row n (where 1 is on row 0, 1 on row 1, 3 and 4 on row 2, etc.). A252745(n) gives the number of nodes on level n whose left child is smaller than the right child, and A252744(n) is an indicator function for those nodes.

(End)

Note that the idea behind maps like this (and the mirror image A005940) admits also using alternative orderings of primes, not just standard magnitude-wise ordering (A000040). For example, A332214 is a similar sequence but with primes rearranged as in A332211, and A332817 is obtained when primes are rearranged as in A108546. - Antti Karttunen, Mar 11 2020

From Lorenzo Sauras Altuzarra, Nov 28 2020: (Start)

This sequence is generated from A228351 by applying the following procedure: 1) eliminate the compositions that end in one unless the first one, 2) subtract one unit from every component, 3) replace every tuple [t_1, ..., t_r] by Product_{k=1..r} A000040(k)^(t_k) (see the examples).

Is it true that a(n) = A337909(n+1) if and only if a(n+1) is not a term of A161992?

Does this permutation have any other cycle apart from (1), (2) and (6, 9, 16, 7)? (End)

Antti Karttunen, Table of n, a(n) for n = 0..8192

Index entries for sequences related to binary expansion of n

Index entries for sequences that are permutations of the natural numbers

Index entries for sequences computed from indices in prime factorization

For n >= 1, a(2n) is even, a(2n+1) is odd. a(2^k) = 2^(k+1), for all k >= 0.

From Antti Karttunen, Jun 20 2014: (Start)

a(0) = 1, a(1) = 2, a(2n) = 2*a(n), a(2n+1) = A003961(a(n)).

As a more general observation about the parity, we have:

For n >= 1, A007814(a(n)) = A135523(n) = A007814(n) + A209229(n). [This permutation preserves the 2-adic valuation of n, except when n is a power of two, in which cases that value is incremented by one.]

For n >= 1, A055396(a(n)) = A091090(n) = A007814(n+1) + 1 - A036987(n).

For n >= 1, a(A000225(n)) = A000040(n).

(End)

From Antti Karttunen, Oct 11 2014: (Start)

As a composition of related permutations:

a(n) = A005940(1+A054429(n)).

a(n) = A064216(A245612(n))

a(n) = A246681(A246378(n)).

Also, for all n >= 0, it holds that:

A161511(n) = A243503(a(n)).

A243499(n) = A243504(a(n)).

(End)

More linking identities from Antti Karttunen, Dec 30 2017: (Start)

A046523(a(n)) = A278531(n). [See also A286531.]

A278224(a(n)) = A285713(n). [Another filter-sequence.]

A048675(a(n)) = A135529(n) seems to hold for n >= 1.

A250245(a(n)) = A252755(n).

A252742(a(n)) = A252744(n).

A245611(a(n)) = A253891(n).

A249824(a(n)) = A275716(n).

A292263(a(n)) = A292264(n). [A292944(n) + A292264(n) = n.]

--

A292383(a(n)) = A292274(n).

A292385(a(n)) = A292271(n). [A292271(n) + A292274(n) = n.]

--

A292941(a(n)) = A292942(n).

A292943(a(n)) = A292944(n).

A292945(a(n)) = A292946(n). [A292942(n) + A292944(n) + A292946(n) = n.]

--

A292253(a(n)) = A292254(n).

A292255(a(n)) = A292256(n). [A292944(n) + A292254(n) + A292256(n) = n.]

--

A279339(a(n)) = A279342(n).

a(A071574(n)) = A269847(n).

a(A279341(n)) = A279338(n).

a(A252756(n)) = A250246(n).

(1+A008836(a(n)))/2 = A059448(n).

(End)

For n=3, whose binary representation is "11", we have A000120(3)=2, with A163510(3,1) = A163510(3,2) = 0, thus a(3) = p(2) * p(1)^0 * p(2)^0 = 3*1*1 = 3.

For n=9, "1001" in binary, we have A000120(9)=2, with A163510(9,1) = 0 and A163510(9,2) = 2, thus a(9) = p(2) * p(1)^0 * p(2)^2 = 3*1*9 = 27.

For n=10, "1010" in binary, we have A000120(10)=2, with A163510(10,1) = 1 and A163510(10,2) = 1, thus a(10) = p(2) * p(1)^1 * p(2)^1 = 3*2*3 = 18.

For n=15, "1111" in binary, we have A000120(15)=4, with A163510(15,1) = A163510(15,2) = A163510(15,3) = A163510(15,4) = 0, thus a(15) = p(4) * p(1)^0 * p(2)^0 * p(3)^0 * p(4)^0 = 7*1*1*1*1 = 7.

[1], [2], [1,1], [3], [1,2], [2,1] ... -> [1], [2], [3], [1,2], ... -> [0], [1], [2], [0,1], ... -> 2^0, 2^1, 2^2, 2^0*3^1, ... = 1, 2, 4, 3, ... - Lorenzo Sauras Altuzarra, Nov 28 2020

f[n_] := Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]]; {1}~Join~

Table[Function[t, Prime[t] Product[Prime[m]^(f[n][[m]]), {m, t}]][DigitCount[n, 2, 1]], {n, 120}] (* Michael De Vlieger, Jul 25 2016 *)

(Scheme, with memoizing definec-macro from Antti Karttunen's IntSeq-library)

;; Version based on given recurrence:

(definec (A163511 n) (cond ((<= n 1) (+ n 1)) ((even? n) (* 2 (A163511 (/ n 2)))) (else (A003961 (A163511 (/ (- n 1) 2))))))

;; Version based on Quet's original formula:

(define (A163511 n) (if (zero? n) 1 (let ((w (A000120 n))) (let loop ((p (A000040 w)) (m w)) (cond ((zero? m) p) (else (loop (* p (expt (A000040 m) (A163510 (+ (A000788 (- n 1)) m)))) (- m 1))))))))

;; Antti Karttunen, Jun 20 2014

Inverse: A243071.

Cf. A000040, A000120, A000225, A000788, A003961, A005940, A007814, A054429, A055396, A064216, A135523, A161992, A163510, A245605, A245612, A246375, A246378, A246681, A161511, A228351, A243499, A243503, A243504, A269854, A280873, A285727, A293437, A337909.

Cf. A332214, A332817 (variants).

Sequence in context: A182944 A269385 A252755 * A332817 A332214 A285322

Adjacent sequences: A163508 A163509 A163510 * A163512 A163513 A163514

base,nonn,look

Leroy Quet, Jul 29 2009

More terms computed and examples added by Antti Karttunen, Jun 20 2014

approved