A285714 -id:A285714 - cp's OEIS frontend

A244154 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = A254049(a(n)), a(2n+1) = 3*a(n)-1; composition of A048673 and A005940.

1, 2, 3, 5, 4, 8, 13, 14, 6, 11, 18, 23, 25, 38, 63, 41, 7, 17, 28, 32, 39, 53, 88, 68, 61, 74, 123, 113, 172, 188, 313, 122, 9, 20, 33, 50, 46, 83, 138, 95, 72, 116, 193, 158, 270, 263, 438, 203, 85, 182, 303, 221, 424, 368, 613, 338, 666, 515, 858, 563, 1201, 938, 1563, 365, 10, 26, 43, 59, 60

Offset: 0

Antti Karttunen, Jun 27 2014

Note the indexing: the domain starts from 0, while the range excludes zero.
From Antti Karttunen, May 30 2017: (Start)
This sequence can be represented as a binary tree. Each left hand child is obtained by applying A254049(n) when the parent contains n, and each right hand child is obtained by applying A016789(n-1) (i.e., multiply by 3, subtract one) to the parent's contents:
                                     1
                                     |
                  ...................2...................
                 3                                       5
       4......../ \........8                  13......../ \........14
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   6       11         18       23         25       38         63       41
  7 17   28  32     39  53   88  68     61  74  123  113   172  188  313 122
etc.
This is a mirror image of the tree depicted in A245612.
(End)

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

Inverse: A244153.

Cf. A005940, A048673, A054429, A243065-A243066, A243505-A243506, A245608, A245610, A245612, A016789, A254049, A285712, A285714, A286613.

(define (A244154 n) (A048673 (A005940 (+ 1 n))))
;; Implementing a new recurrence, with memoization-macro definec:
(definec (A244154 n) (cond ((<= n 1) (+ 1 n)) ((even? n) (A254049 (A244154 (/ n 2)))) (else (+ -1 (* 3 (A244154 (/ (- n 1) 2))))))) ;; Antti Karttunen, May 30 2017

a(n) = A048673(A005940(n+1)).
From Antti Karttunen, May 30 2017: (Start)
a(0) = 1, a(1) = 2, a(2n) = A254049(a(n)), a(2n+1) = 3*a(n)-1.
a(n) = A245612(A054429(n)).
(End)

A285712 a(1) = 0, and for n > 1, if n = 3k-1, then a(n) = k, otherwise a(n) = (A064216(n)+1)/2.

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 6, 3, 7, 9, 4, 10, 5, 5, 12, 15, 6, 8, 16, 7, 19, 21, 8, 22, 13, 9, 24, 11, 10, 27, 30, 11, 17, 31, 12, 34, 36, 13, 18, 37, 14, 40, 20, 15, 42, 28, 16, 26, 45, 17, 49, 51, 18, 52, 54, 19, 55, 29, 20, 33, 25, 21, 14, 57, 22, 64, 43, 23, 66, 69, 24, 39, 35, 25, 70, 75, 26, 44, 76, 27, 48, 79, 28, 82, 61, 29, 84, 23, 30, 87, 90, 31, 47, 46, 32

Offset: 1

Antti Karttunen, Apr 25 2017

nonn

For n >= 2, a(n) gives the contents of the parent node of the node containing n in binary trees like A245612.

Every positive integer greater than one occurs exactly twice in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A002264, A048673, A064216, A245612, A285713, A285714.

a[n_] := a[n] = Which[n == 1, 0, Mod[n, 3] == 2, Ceiling[n/3], True, (Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1] + 1)/2]; Array[a, 95] (* Michael De Vlieger, Sep 22 2017 *)

(define (A285712 n) (cond ((<= n 1) (- n 1)) ((= 2 (modulo n 3)) (A002264 (+ 1 n))) (else (/ (+ 1 (A064216 n)) 2))))

a(1) = 0, and for n > 1, if n = 3*k-1, then a(n) = k, otherwise a(n) = (A064216(n)+1)/2.

a(n) = (n+1)/3 + (3*A064216(n) - 2*n + 1)*( (n+1)^2 mod 3 )/6, for n>1. - Ammar Khatab, Sep 21 2020

A291763 Binary encoding of 2-digits in ternary representation of A245612(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 3, 0, 1, 8, 1, 6, 5, 4, 1, 2, 1, 10, 23, 16, 1, 2, 13, 10, 9, 2, 7, 0, 1, 0, 3, 2, 1, 70, 21, 24, 45, 4, 33, 52, 1, 36, 3, 20, 25, 10, 21, 0, 17, 18, 5, 0, 13, 16, 3, 12, 1, 8, 1, 4, 5, 2, 5, 0, 1, 32, 139, 74, 41, 208, 49, 0, 89, 108, 11, 130, 65, 18, 103, 4, 1, 8, 73, 4, 5, 112, 41, 16, 49, 72, 19, 38, 41, 20, 1, 8, 33, 0, 35, 86, 9, 38

Offset: 0

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A245612, A285712, A285714, A285715, A285716, A289814, A291759, A291760.

a(n) = A289814(A245612(n)).

A285715 a(n) = A000120(A245611(n)).

Original entry on oeis.org

0, 1, 2, 3, 1, 4, 5, 2, 6, 7, 3, 8, 2, 1, 9, 10, 4, 3, 11, 5, 12, 13, 2, 14, 3, 6, 15, 4, 7, 16, 17, 3, 5, 18, 8, 19, 20, 2, 4, 21, 1, 22, 6, 9, 23, 5, 10, 7, 24, 4, 25, 26, 3, 27, 28, 11, 29, 8, 5, 6, 4, 12, 2, 30, 13, 31, 7, 2, 32, 33, 14, 5, 9, 3, 34, 35, 6, 10, 36, 15, 8, 37, 4, 38, 5, 7, 39, 3, 16, 40, 41, 17, 11, 6, 3, 42, 43, 5, 44, 45, 18, 9, 12, 8, 7

Offset: 1

Antti Karttunen, Apr 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000120, A245611, A285712, A285714, A285716.

Cf. A007051 (positions of 0 and 1's).

;; First implementation uses memoization-macro definec:
(definec (A285715 n) (if (<= n 2) (- n 1) (+ (if (= 2 (modulo n 3)) 0 1) (A285715 (A285712 n)))))
(define (A285715 n) (A000120 (A245611 n)))

a(1) = 0, a(2) = 1, for n > 2, a(n) = a(A285712(n)) + [n <> 2 mod 3]. (Where [] is Iverson bracket, giving here 1 if n is of the form 3k or 3k+1, and 0 if it is of the form 3k+2.)

a(n) = A000120(A245611(n)).

A285716 a(n) = A080791(A245611(n)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 2, 0, 0, 1, 0, 1, 2, 1, 1, 1, 2, 0, 1, 0, 1, 3, 0, 0, 1, 1, 1, 2, 0, 0, 2, 1, 0, 1, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 0, 1, 1, 1, 3, 0, 0, 2, 0, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 0, 3, 0, 0, 2, 0, 1, 1, 0

Offset: 1

Antti Karttunen, Apr 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

One less than A091304 after the initial term.
Cf. A000120, A080791, A244153, A245611, A278223, A285712, A285714, A285715.
Cf. A006254 (gives the positions of zeros after initial a(1)=0.)

a[n_] := PrimeOmega[2*n - 1] - 1; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

;; First implementation uses memoization-macro definec:
(definec (A285716 n) (if (<= n 2) 0 (+ (if (= 2 (modulo n 3)) 1 0) (A285716 (A285712 n)))))
(define (A285716 n) (A080791 (A245611 n)))

a(1) = 0, a(2) = 1, for n > 2, a(n) = a(A285712(n)) + [n == 2 mod 3]. (Where [] is Iverson bracket, giving here 1 only if n is of the form 3k+2, and 0 otherwise.)
a(n) = A080791(A245611(n)).
For all n >= 2,  a(n) = A091304(n)-1 = A000120(A244153(n))-1. - Antti Karttunen, May 31 2017

cp's OEIS Frontend

A244154 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = A254049(a(n)), a(2n+1) = 3*a(n)-1; composition of A048673 and A005940.

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A285712 a(1) = 0, and for n > 1, if n = 3k-1, then a(n) = k, otherwise a(n) = (A064216(n)+1)/2.

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A291763 Binary encoding of 2-digits in ternary representation of A245612(n).

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A285715 a(n) = A000120(A245611(n)).

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A285716 a(n) = A080791(A245611(n)).

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