A249745 -id:A249745 - cp's OEIS frontend

A254053 Square array: A(row,col) = 2^(row-1) * ((2*A249745(col))-1) = A064216(A254051(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

1, 3, 2, 5, 6, 4, 7, 10, 12, 8, 11, 14, 20, 24, 16, 13, 22, 28, 40, 48, 32, 17, 26, 44, 56, 80, 96, 64, 19, 34, 52, 88, 112, 160, 192, 128, 9, 38, 68, 104, 176, 224, 320, 384, 256, 23, 18, 76, 136, 208, 352, 448, 640, 768, 512, 29, 46, 36, 152, 272, 416, 704, 896, 1280, 1536, 1024, 15, 58, 92, 72, 304, 544, 832, 1408, 1792, 2560, 3072, 2048, 31, 30

Offset: 1

Antti Karttunen, Jan 24 2015

Shares with A135764 and A253551 the property that A001511(n) = k for all terms n on row k and when going downward in each column, terms grow by doubling.

			The top left corner of the array:
   1,  3,  5,   7,  11,  13,  17,  19,   9,  23,  29,  15,  31,  37,  41,  43,
   2,  6, 10,  14,  22,  26,  34,  38,  18,  46,  58,  30,  62,  74,  82,  86,
   4, 12, 20,  28,  44,  52,  68,  76,  36,  92, 116,  60, 124, 148, 164, 172,
   8, 24, 40,  56,  88, 104, 136, 152,  72, 184, 232, 120, 248, 296, 328, 344,
  16, 48, 80, 112, 176, 208, 272, 304, 144, 368, 464, 240, 496, 592, 656, 688,
...

Inverse: A254054.
Similar or related permutations: A135764, A253551, A064216, A254051.
Cf. A000079, A001511, A002260, A004736, A064989, A135765, A249745, A254049, A254050.

A(row,col) = A135764(row, A249745(col)). [Is otherwise the same array as A135764, but the column positions have been permuted by A249745.]
A(row,col) = 2^(row-1) * ((2*A249745(col))-1) = 2^(row-1) * A254050(col). [The above expands to this.]
a(n) = A064989(A135765(n)).
As a composition of other permutations:
a(n) = A064216(A254051(n)). [As an array: A(row,col) = A064216(A254051(row,col)).]

A254050 Permutation of odd numbers: a(n) = (2*(A249745(n))) - 1 = A064989(A007310(n)).

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 9, 23, 29, 15, 31, 37, 41, 43, 25, 47, 21, 53, 59, 33, 61, 67, 71, 35, 73, 79, 39, 83, 55, 51, 89, 97, 101, 103, 107, 109, 57, 65, 49, 27, 113, 127, 85, 131, 137, 77, 69, 139, 149, 87, 151, 95, 157, 163, 121, 167, 45, 173, 179, 93, 91, 181, 191, 193, 197, 115, 111, 119, 199, 123

Offset: 1

Antti Karttunen, Jan 26 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191

Cf. A007310, A064989, A249745, A254049, A254051, A254053.

a(n) = (2*(A249745(n))) - 1.

a(n) = A064989(A007310(n)).

A254118 Permutation of natural numbers: a(n) = A249745(1+A254103(n)) - 1.

Original entry on oeis.org

1, 2, 3, 6, 5, 4, 8, 20, 11, 7, 9, 33, 18, 23, 14, 13, 30, 36, 21, 44, 10, 29, 15, 55, 53, 28, 16, 74, 39, 41, 12, 179, 90, 96, 50, 114, 24, 42, 35, 92, 69, 47, 19, 86, 25, 51, 26, 236, 153, 110, 81, 101, 22, 45, 48, 221, 113, 119, 56, 77, 65, 38, 17, 546, 182

Offset: 1

Antti Karttunen, Feb 05 2015

Antti Karttunen, Table of n, a(n) for n = 1..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A254117.
Other related permutations: A254116, A249745, A254103 (compare to the scatterplot of this one).
Cf. A254120 (= a(2^n)).

default(primelimit, 2^30);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
A254103(n) = { if(0==n,0,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2)); };
A254116(n) = A064216(A254103(n));
A254118(n) = (A254116(n+n+1)-1)/2;
for(n=1, 8191, write("b254118.txt", n, " ", A254118(n)));
(Scheme, two versions)
(define (A254118 n) (+ -1 (A249745 (+ 1 (A254103 n)))))
(define (A254118 n) (/ (+ -1 (A254116 (+ 1 n n))) 2))

from sympy import factorint, prevprime, floor
from operator import mul
from functools import reduce
def a064216(n):
    f=factorint(2*n - 1)
    return 1 if n==1 else reduce(mul, [prevprime(i)**f[i] for i in f])
def a254103(n):
    if n==0: return 0
    if n%2==0: return 3*a254103(n//2) - 1
    else: return floor((3*(1 + a254103((n - 1)/2)))//2)
def a254116(n): return a064216(a254103(n))
def a(n): return (a254116(2*n + 1) - 1)//2
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

a(n) = A249745(1+A254103(n)) - 1.

a(n) = (A254116((2*n)+1)-1) / 2. [Obtained also from the odd bisection of A254116.]

A064989 Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 10, 7, 19, 2, 9, 11, 8, 5, 23, 6, 29, 1, 14, 13, 15, 4, 31, 17, 22, 3, 37, 10, 41, 7, 12, 19, 43, 2, 25, 9, 26, 11, 47, 8, 21, 5, 34, 23, 53, 6, 59, 29, 20, 1, 33, 14, 61, 13, 38, 15, 67, 4, 71, 31, 18, 17, 35, 22, 73, 3, 16

Offset: 1

Vladeta Jovovic, Oct 30 2001

From Antti Karttunen, May 12 2014: (Start)
a(A003961(n)) = n for all n. [This is a left inverse function for the injection A003961.]
Bisections are A064216 (the terms at odd indices) and A064989 itself (the terms at even indices), i.e., a(2n) = a(n) for all n.
(End)
From Antti Karttunen, Dec 18-21 2014: (Start)
When n represents an unordered integer partition via the indices of primes present in its prime factorization (for n >= 2, n corresponds to the partition given as the n-th row of A112798) this operation subtracts one from each part. If n is of the form 2^k (a partition having just k 1's as its parts) the result is an empty partition (which is encoded by 1, having an "empty" factorization).
For all odd numbers n >= 3, a(n) tells which number is located immediately above n in square array A246278. Cf. also A246277.
(End)
Alternatively, if numbers are represented as the multiset of indices of prime factors with multiplicity, this operation subtracts 1 from each element and discards the 0's. - M. F. Hasler, Dec 29 2014

			a(20) = a(2^2*5) = a(2^2)*a(5) = prevprime(5) = 3.

Harry J. Smith (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A064216 (odd bisection), A003961 (inverse), A151799.
Cf. A000040, A008578, A027746, A032742, A049084, A052126, A055396, A061395, A112798, A249817, A249818.
Other sequences whose definition involve or are some other way related with this sequence: A105560, A108951, A118306, A122111, A156552, A163511, A200746, A241909, A243070, A243071, A243072, A243073, A244319, A245605, A245607, A246165, A246266, A246268, A246277, A246278, A246361, A246362, A246371, A246372, A246373, A246374, A246376, A246380, A246675, A246682, A249745, A250470.
Similar prime-shifts towards smaller numbers: A252461, A252462, A252463.

a064989 1 = 1
a064989 n = product $ map (a008578 . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A064989 n) (if (= 1 n) n (apply * (map (lambda (k) (if (zero? k) 1 (A000040 k))) (map -1+ (map A049084 (factor n)))))))
;; Antti Karttunen, May 12 2014
(definec (A064989 n) (if (= 1 n) n (* (A008578 (A055396 n)) (A064989 (A032742 n))))) ;; One based on given recurrence and utilizing memoizing definec-macro.
(definec (A064989 n) (cond ((= 1 n) n) ((even? n) (A064989 (/ n 2))) (else (A163511 (/ (- (A243071 n) 1) 2))))) ;; Corresponds to one of the alternative formulas, but is very unpractical way to compute this sequence. - Antti Karttunen, Dec 18 2014

q:= proc(p) prevprime(p) end proc: q(2):= 1:
[seq(mul(q(f[1])^f[2], f = ifactors(n)[2]), n = 1 .. 1000)]; # Robert Israel, Dec 21 2014

Table[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n, {n, 81}] (* Michael De Vlieger, Jan 04 2016 *)

{ for (n=1, 1000, f=factor(n)~; a=1; j=1; if (n>1 && f[1, 1]==2, j=2); for (i=j, length(f), a*=precprime(f[1, i] - 1)^f[2, i]); write("b064989.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 02 2009

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((p=f[i,1]) % 2, f[i,1] = precprime(p-1), f[i,1] = 1);); factorback(f);} \\ Michel Marcus, Dec 18 2014

A064989(n)=factorback(Mat(apply(t->[max(precprime(t[1]-1),1),t[2]],Vec(factor(n)~))~)) \\ M. F. Hasler, Dec 29 2014

from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

from math import prod
from sympy import prevprime, factorint
def A064989(n): return prod(prevprime(p)**e for p, e in  factorint(n>>(~n&n-1).bit_length()).items()) # Chai Wah Wu, Jan 05 2023

From Antti Karttunen, Dec 18 2014: (Start)
If n = product A000040(k)^e(k) then a(n) = product A008578(k)^e(k) [where A000040(n) gives the n-th prime, and A008578(n) gives 1 for 1 and otherwise the (n-1)-th prime].
a(1) = 1; for n > 1, a(n) = A008578(A055396(n)) * a(A032742(n)). [Above formula represented as a recurrence. Cf. A252461.]
a(1) = 1; for n > 1, a(n) = A008578(A061395(n)) * a(A052126(n)). [Compare to the formula of A252462.]
This prime-shift operation is used in the definitions of many other sequences, thus it can be expressed in many alternative ways:
a(n) = A200746(n) / n.
a(n) = A242424(n) / A105560(n).
a(n) = A122111(A122111(n)/A105560(n)) = A122111(A052126(A122111(n))). [In A112798-partition context: conjugate, remove the largest part (the largest prime factor), and conjugate again.]
a(1) = 1; for n > 1, a(2n) = a(n), a(2n+1) = A163511((A243071(2n+1)-1) / 2).
a(n) = A249818(A250470(A249817(n))). [A250470 is an analogous operation for "going one step up" in the square array A083221 (A083140).]
(End)
Product_{k=1..n} a(k) = n! / A307035(n). - Vaclav Kotesovec, Mar 21 2019
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) = 0.220703928... , where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Nov 18 2022

A048673 Permutation of natural numbers: a(n) = (A003961(n)+1) / 2 [where A003961(n) shifts the prime factorization of n one step towards larger primes].

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 6, 14, 13, 11, 7, 23, 9, 17, 18, 41, 10, 38, 12, 32, 28, 20, 15, 68, 25, 26, 63, 50, 16, 53, 19, 122, 33, 29, 39, 113, 21, 35, 43, 95, 22, 83, 24, 59, 88, 44, 27, 203, 61, 74, 48, 77, 30, 188, 46, 149, 58, 47, 31, 158, 34, 56, 138, 365, 60, 98, 36, 86, 73

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

Inverse of sequence A064216 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001
From Antti Karttunen, Dec 20 2014: (Start)
Permutation of natural numbers obtained by replacing each prime divisor of n with the next prime and mapping the generated odd numbers back to all natural numbers by adding one and then halving.
Note: there is a 7-cycle almost right in the beginning: (6 8 14 17 10 11 7). (See also comments at A249821. This 7-cycle is endlessly copied in permutations like A250249/A250250.)
The only 3-cycle in range 1 .. 402653184 is (2821 3460 5639).
For 1- and 2-cycles, see A245449.
(End)
The first 5-cycle is (1410, 2783, 2451, 2703, 2803). - Robert Israel, Jan 15 2015
From Michel Marcus, Aug 09 2020: (Start)
(5194, 5356, 6149, 8186, 10709), (46048, 51339, 87915, 102673, 137205) and (175811, 200924, 226175, 246397, 267838) are other 5-cycles.
(10242, 20479, 21413, 29245, 30275, 40354, 48241) is another 7-cycle. (End)
From Antti Karttunen, Feb 10 2021: (Start)
Somewhat artificially, also this permutation can be represented as a binary tree. Each child to the left is obtained by multiplying the parent by 3 and subtracting one, while each child to the right is obtained by applying A253888 to the parent:
                                       1
                                       |
                    ................../ \..................
                   2                                       3
         5......../ \........4                   8......../ \........6
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    14       13         11       7          23       9          17       18
  41 10    38  12     32  28   20 15      68  25   26 63      50  16   53  19
etc.
Each node's (> 1) parent can be obtained with A253889. Sequences A292243, A292244, A292245 and A292246 are constructed from the residues (mod 3) of the vertices encountered on the path from n to the root (1).
(End)

			For n = 6, as 6 = 2 * 3 = prime(1) * prime(2), we have a(6) = ((prime(1+1) * prime(2+1))+1) / 2 = ((3 * 5)+1)/2 = 8.
For n = 12, as 12 = 2^2 * 3, we have a(12) = ((3^2 * 5) + 1)/2 = 23.

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

Inverse: A064216.
Row 1 of A251722, Row 2 of A249822.
One more than A108228, half the terms of A243501.
Fixed points: A048674.
Positions of records: A029744, their values: A246360 (= A007051 interleaved with A057198).
Positions of subrecords: A247283, their values: A247284.
Cf. A246351 (Numbers n such that a(n) < n.)
Cf. A246352 (Numbers n such that a(n) >= n.)
Cf. A246281 (Numbers n such that a(n) <= n.)
Cf. A246282 (Numbers n such that a(n) > n.), A252742 (their char. function)
Cf. A246261 (Numbers n for which a(n) is odd.)
Cf. A246263 (Numbers n for which a(n) is even.)
Cf. A246260 (a(n) reduced modulo 2), A341345 (modulo 3), A341346, A292251 (3-adic valuation), A292252.
Cf. A246342 (Iterates starting from n=12.)
Cf. A246344 (Iterates starting from n=16.)
Cf. also A003602, A003961 (A045965), A108951, A151800, A245449, A249735, A249821, A250471, A250249, A250250.
Cf. A245447 (This permutation "squared", a(a(n)).)
Other permutations whose formulas refer to this sequence: A122111, A243062, A243066, A243500, A243506, A244154, A244319, A245605, A245608, A245610, A245612, A245708, A246265, A246267, A246268, A246363, A249745, A249824, A249826, and also A183209, A254103 that are somewhat similar.
Cf. also prime-shift based binary trees A005940, A163511, A245612 and A244154.
Cf. A253888, A253889, A292243, A292244, A292245 and A292246 for other derived sequences.
Cf. A323893 (Dirichlet inverse), A323894 (sum with it), A336840 (inverse Möbius transform).

a048673 = (`div` 2) . (+ 1) . a045965
-- Reinhard Zumkeller, Jul 12 2012

f:= proc(n)
local F,q,t;
  F:= ifactors(n)[2];
  (1 + mul(nextprime(t[1])^t[2], t = F))/2
end proc:
seq(f(n),n=1..1000); # Robert Israel, Jan 15 2015

Table[(Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n, {n, 69}] (* Michael De Vlieger, Dec 18 2014, revised Mar 17 2016 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2; \\ Antti Karttunen, Dec 20 2014

A048673(n) = if(1==n,n,if(n%2,A253888(A048673((n-1)/2)),(3*A048673(n/2))-1)); \\ (Not practical, but demonstrates the construction as a binary tree). - Antti Karttunen, Feb 10 2021

from sympy import factorint, nextprime, prod
def a(n):
    f = factorint(n)
    return 1 if n==1 else (1 + prod(nextprime(i)**f[i] for i in f))//2 # Indranil Ghosh, May 09 2017

(define (A048673 n) (/ (+ 1 (A003961 n)) 2)) ;; Antti Karttunen, Dec 20 2014

From Antti Karttunen, Dec 20 2014: (Start)
a(1) = 1; for n>1: If n = product_{k>=1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k>=1} (p_{k+1})^(c_k)).
a(n) = (A003961(n)+1) / 2.
a(n) = floor((A045965(n)+1)/2).
Other identities. For all n >= 1:
a(n) = A108228(n)+1.
a(n) = A243501(n)/2.
A108951(n) = A181812(a(n)).
a(A246263(A246268(n))) = 2*n.
As a composition of other permutations involving prime-shift operations:
a(n) = A243506(A122111(n)).
a(n) = A243066(A241909(n)).
a(n) = A241909(A243062(n)).
a(n) = A244154(A156552(n)).
a(n) = A245610(A244319(n)).
a(n) = A227413(A246363(n)).
a(n) = A245612(A243071(n)).
a(n) = A245608(A245605(n)).
a(n) = A245610(A244319(n)).
a(n) = A249745(A249824(n)).
For n >= 2, a(n) = A245708(1+A245605(n-1)).
(End)
From Antti Karttunen, Jan 17 2015: (Start)
We also have the following identities:
a(2n) = 3*a(n) - 1. [Thus a(2n+1) = 0 or 1 when reduced modulo 3. See A341346]
a(3n) = 5*a(n) - 2.
a(4n) = 9*a(n) - 4.
a(5n) = 7*a(n) - 3.
a(6n) = 15*a(n) - 7.
a(7n) = 11*a(n) - 5.
a(8n) = 27*a(n) - 13.
a(9n) = 25*a(n) - 12.
and in general:
a(x*y) = (A003961(x) * a(y)) - a(x) + 1, for all x, y >= 1.
(End)
From Antti Karttunen, Feb 10 2021: (Start)
For n > 1, a(2n) = A016789(a(n)-1), a(2n+1) = A253888(a(n)).
a(2^n) = A007051(n) for all n >= 0. [A property shared with A183209 and A254103].
(End)
a(n) = A003602(A003961(n)). - Antti Karttunen, Apr 20 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/4) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 1.0319981... , where nextprime is A151800. - Amiram Eldar, Jan 18 2023

New name and crossrefs to derived sequences added by Antti Karttunen, Dec 20 2014

A003602 Kimberling's paraphrases: if n = (2k-1)*2^m then a(n) = k.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mira Bernstein

Fractal sequence obtained from powers of 2.
k occurs at (2*k-1)*A000079(m), m >= 0. - Robert G. Wilson v, May 23 2006
Sequence is T^(oo)(1) where T is acting on a word w = w(1)w(2)..w(m) as follows: T(w) = "1"w(1)"2"w(2)"3"(...)"m"w(m)"m+1". For instance T(ab) = 1a2b3. Thus T(1) = 112, T(T(1)) = 1121324, T(T(T(1))) = 112132415362748. - Benoit Cloitre, Mar 02 2009
Note that iterating the post-numbering operator U(w) = w(1) 1 w(2) 2 w(3) 3... produces the same limit sequence except with an additional "1" prepended, i.e., 1,1,1,2,1,3,2,4,... - Glen Whitney, Aug 30 2023
In the binary expansion of n, first swallow all zeros from the right, then add 1, and swallow the now-appearing 0 bit as well. - Ralf Stephan, Aug 22 2013
Although A264646 and this sequence initially agree in their digit-streams, they differ after 48 digits. - N. J. A. Sloane, Nov 20 2015
"[This is a] fractal because we get the same sequence after we delete from it the first appearance of all positive integers" - see Cobeli and Zaharescu link. - Robert G. Wilson v, Jun 03 2018
From Peter Munn, Jun 16 2022: (Start)
The sequence is the list of positive integers interleaved with the sequence itself. Provided the offset is suitable (which is the case here) a term of such a self-interleaved sequence is determined by the odd part of its index. Putting some of the formulas given here into words, a(n) is the position of the odd part of n in the list of odd numbers.
Applying the interleaving transform again, we get A110963.
(End)
Omitting all 1's leaves A131987 + 1. - David James Sycamore, Jul 26 2022
a(n) is also the smallest positive number not among the terms between a(a(n-1)) and a(n-1) inclusive (with a(0)=1 prepended). - Neal Gersh Tolunsky, Mar 07 2023

			From _Peter Munn_, Jun 14 2022: (Start)
Start of table showing the interleaving with the positive integers:
   n  a(n)  (n+1)/2  a(n/2)
   1    1      1
   2    1               1
   3    2      2
   4    1               1
   5    3      3
   6    2               2
   7    4      4
   8    1               1
   9    5      5
  10    3               3
  11    6      6
  12    2               2
(End)

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Cristian Cobeli and Alexandru Zaharescu, Promenade around Pascal Triangle - Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, 73-98.
J.-P. Delahaye, La marelle arithmétique, Pour la Science, No. 360, October 2007. In French.
Dale Gerdemann, Plotting Adjacent Points in A003602, Kimberling's Paraphrase, YouTube Video, 2015.
Dale Gerdemann, Plotting Adjacent Terms of A003602 Modulo Increasing Powers of 2, YouTube Video, 2015.
Douglas E. Iannucci and Urban Larsson, Game values of arithmetic functions, arXiv:2101.07608 [math.NT], 2021.
Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv preprint arXiv:1608.00862 [math.GM], 2016.
Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
Clark Kimberling, Fractal sequences
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Matty van-Son, Palindromic sequences of the Markov spectrum, arXiv:1804.10802 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Odd Part
Index entries for sequences related to binary expansion of n

a(n) is the index of the column in A135764 where n appears (see also A054582).
Cf. A000079, A000265, A001511, A003603, A003961, A014577 (with offset 1, reduction mod 2), A025480, A035528, A048673, A101279, A110963, A117303, A126760, A181988, A220466, A249745, A253887, A337821 (2-adic valuation).
Cf. also A349134 (Dirichlet inverse), A349135 (sum with it), A349136 (Möbius transform), A349431, A349371 (inverse Möbius transform).
Cf. A065120, A131987.
Cf. A264646.
Cf. A123390, A118319.

a003602 = (`div` 2) . (+ 1) . a000265
-- Reinhard Zumkeller, Feb 16 2012, Oct 14 2010

import Data.List (transpose)
a003602 = flip div 2 . (+ 1) . a000265
a003602_list = concat $ transpose [[1..], a003602_list]
-- Reinhard Zumkeller, Aug 09 2013, May 23 2013

A003602:=proc(n) options remember: if n mod 2 = 1 then RETURN((n+1)/2) else RETURN(procname(n/2)) fi: end proc:
seq(A003602(n), n=1..83); # Pab Ter
nmax := 83: for m from 0 to ceil(simplify(log[2](nmax))) do for k from 1 to ceil(nmax/(m+2)) do a((2*k-1)*2^m) := k od: od: seq(a(k), k=1..nmax); # Johannes W. Meijer, Feb 04 2013
A003602 := proc(n)
    a := 1;
    for p in ifactors(n)[2] do
        if op(1,p) > 2 then
            a := a*op(1,p)^op(2,p) ;
        end if;
    end do  :
    (a+1)/2 ;
end proc: # R. J. Mathar, May 19 2016

a[n_] := Block[{m = n}, While[ EvenQ@m, m /= 2]; (m + 1)/2]; Array[a, 84] (* or *)
a[1] = 1; a[n_] := a[n] = If[OddQ@n, (n + 1)/2, a[n/2]]; Array[a, 84] (* Robert G. Wilson v, May 23 2006 *)
a[n_] := Ceiling[NestWhile[Floor[#/2] &, n, EvenQ]/2]; Array[a, 84] (* Birkas Gyorgy, Apr 05 2011 *)
a003602 = {1}; max = 7; Do[b = {}; Do[AppendTo[b, {k, a003602[[k]]}], {k, Length[a003602]}]; a003602 = Flatten[b], {n, 2, max}]; a003602 (* L. Edson Jeffery, Nov 21 2015 *)

A003602(n)=(n/2^valuation(n,2)+1)/2; /* Joerg Arndt, Apr 06 2011 */

import math
def a(n): return (n/2**int(math.log(n - (n & n - 1), 2)) + 1)/2 # Indranil Ghosh, Apr 24 2017

def A003602(n): return (n>>(n&-n).bit_length())+1 # Chai Wah Wu, Jul 08 2022

(define (A003602 n) (let loop ((n n)) (if (even? n) (loop (/ n 2)) (/ (+ 1 n) 2)))) ;; Antti Karttunen, Feb 04 2015

a(n) = (A000265(n) + 1)/2.
a((2*k-1)*2^m) = k, for m >= 0 and k >= 1. - Robert G. Wilson v, May 23 2006
Inverse Weigh transform of A035528. - Christian G. Bower
G.f.: 1/x * Sum_{k>=0} x^2^k/(1-2*x^2^(k+1) + x^2^(k+2)). - Ralf Stephan, Jul 24 2003
a(2*n-1) = n and a(2*n) = a(n). - Pab Ter (pabrlos2(AT)yahoo.com), Oct 25 2005
a(A118413(n,k)) = A002024(n,k); = a(A118416(n,k)) = A002260(n,k); a(A014480(n)) = A001511(A014480(n)). - Reinhard Zumkeller, Apr 27 2006
Ordinal transform of A001511. - Franklin T. Adams-Watters, Aug 28 2006
a(n) = A249745(A126760(A003961(n))) = A249745(A253887(A048673(n))). That is, this sequence plays the same role for the numbers in array A135764 as A126760 does for the odd numbers in array A135765. - Antti Karttunen, Feb 04 2015 & Jan 19 2016
G.f. satisfies g(x) = g(x^2) + x/(1-x^2)^2. - Robert Israel, Apr 24 2015
a(n) = A181988(n)/A001511(n). - L. Edson Jeffery, Nov 21 2015
a(n) = A025480(n-1) + 1. - R. J. Mathar, May 19 2016
a(n) = A110963(2n-1) = A349135(4*n). - Antti Karttunen, Apr 18 2022
a(n) = (1 + n)/2, for n odd; a(n) = a(n/2), for n even. - David James Sycamore, Jul 28 2022
a(n) = n/2^A001511(n) + 1/2. - Alan Michael Gómez Calderón, Oct 06 2023
a(n) = A123390(A118319(n)). - Flávio V. Fernandes, Mar 02 2025

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 25 2005

A032766 Numbers that are congruent to 0 or 1 (mod 3).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 102, 103

Offset: 0

Patrick De Geest, May 15 1998

Omitting the initial 0, a(n) is the number of 1's in the n-th row of the triangle in A118111. - Hans Havermann, May 26 2002
Binomial transform is A053220. - Michael Somos, Jul 10 2003
Smallest number of different people in a set of n-1 photographs that satisfies the following conditions: In each photograph there are 3 women, the woman in the middle is the mother of the person on her left and is a sister of the person on her right and the women in the middle of the photographs are all different. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
Partial sums of A000034. - Richard Choulet, Jan 28 2010
Starting with 1 = row sums of triangle A171370. - Gary W. Adamson, Feb 15 2010
a(n) is the set of values for m in which 6k + m can be a perfect square (quadratic residues of 6 including trivial case of 0). - Gary Detlefs, Mar 19 2010
For n >= 2, a(n) is the smallest number with n as an anti-divisor. - Franklin T. Adams-Watters, Oct 28 2011
Sequence is also the maximum number of floors with 3 elevators and n stops in a "Convenient Building". See A196592 and Erich Friedman link below. - Robert Price, May 30 2013
a(n) is also the total number of coins left after packing 4-curves patterns (4c2) into a fountain of coins base n. The total number of 4c2 is A002620 and voids left is A000982. See illustration in links. - Kival Ngaokrajang, Oct 26 2013
Number of partitions of 6n into two even parts. - Wesley Ivan Hurt, Nov 15 2014
Number of partitions of 3n into exactly 2 parts. - Colin Barker, Mar 23 2015
Nonnegative m such that floor(2*m/3) = 2*floor(m/3). - Bruno Berselli, Dec 09 2015
For n >= 3, also the independence number of the n-web graph. - Eric W. Weisstein, Dec 31 2015
Equivalently, nonnegative numbers m for which m*(m+2)/3 and m*(m+5)/6 are integers. - Bruno Berselli, Jul 18 2016
Also the clique covering number of the n-Andrásfai graph for n > 0. - Eric W. Weisstein, Mar 26 2018
Maximum sum of degeneracies over all decompositions of the complete graph of order n+1 into three factors. The extremal decompositions are characterized in the Bickle link below. - Allan Bickle, Dec 21 2021
Also the Hadwiger number of the n-cocktail party graph. - Eric W. Weisstein, Apr 30 2022
The number of integer rectangles with a side of length n+1 and the property: the bisectors of the angles form a square within its limits. - Alexander M. Domashenko, Oct 17 2024
The maximum possible number of 5-cycles in an outerplanar graph on n+4 vertices. - Stephen Bartell, Jul 10 2025

Indranil Ghosh, Table of n, a(n) for n = 0..10000
Allan Bickle, Nordhaus-Gaddum Theorems for k-Decompositions, Congr. Num. 211 (2012) 171-183.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Erich Friedman, Problem of the month November 2009
Z. Füredi, A. Kostochka, M. Stiebitz, R. Skrekovski, and D. West, Nordhaus-Gaddum-type theorems for decompositions into many parts, J. Graph Theory 50 (2005), 273-292.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 282. [Book's website]
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest, Problem #20. [Broken link; Cached copy]
Matroids Matheplanet, Number of d-generator groups of order 2^(d+1) and exponent-p class 2 (in German).
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche, Vol. 76, No. 1 (2021), pp. 277-310, see p. 302.
Kival Ngaokrajang, Illustration of initial terms (U).
Eric Weisstein's World of Mathematics, Andrásfai Graph
Eric Weisstein's World of Mathematics, Clique Covering Number
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Hadwiger Number
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Web Graph
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A006578 (partial sums), A000034 (first differences), A016789 (complement).
Essentially the same: A049624.
Column 1 (the second leftmost) of triangular table A026374.
Column 1 (the leftmost) of square array A191450.
Row 1 of A254051.
Row sums of A171370.
Cf. A066272 for anti-divisors.
Cf. A253888 and A254049 (permutations of this sequence without the initial zero).
Cf. A254103 and A254104 (pair of permutations based on this sequence and its complement).
Cf. A000035, A000217, A000982, A001651, A002620, A002717, A003945.
Cf. A004396, A004526, A007310, A007494, A008619, A030308, A035360.
Cf. A047270, A049615, A052928, A053220, A070893, A084056, A092942.
Cf. A118111, A132463, A133083, A196592, A249745.

a032766 n = div n 2 + n  -- Reinhard Zumkeller, Dec 13 2014
(MIT/GNU Scheme) (define (A032766 n) (+ n (floor->exact (/ n 2)))) ;; Antti Karttunen, Jan 24 2015

&cat[ [n, n+1]: n in [0..100 by 3] ]; // Vincenzo Librandi, Nov 16 2014

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..69); # Zerinvary Lajos, Mar 16 2008
seq(floor(n/2)+n, n=0..69); # Gary Detlefs, Mar 19 2010
select(n->member(n mod 3,{0,1}), [$0..103]); # Peter Luschny, Apr 06 2014

a[n_] := a[n] = 2a[n - 1] - 2a[n - 3] + a[n - 4]; a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 4; Array[a, 60, 0] (* Robert G. Wilson v, Mar 28 2011 *)
Select[Range[0, 200], MemberQ[{0, 1}, Mod[#, 3]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
Flatten[{#,#+1}&/@(3Range[0,40])] (* or *) LinearRecurrence[{1,1,-1}, {0,1,3}, 100] (* or *) With[{nn=110}, Complement[Range[0,nn], Range[2,nn,3]]] (* Harvey P. Dale, Mar 10 2013 *)
CoefficientList[Series[x (1 + 2 x) / ((1 - x) (1 - x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 16 2014 *)
Floor[3 Range[0, 69]/2] (* L. Edson Jeffery, Jan 14 2017 *)
Drop[Range[0,110],{3,-1,3}] (* Harvey P. Dale, Sep 02 2023 *)

PARI
```
{a(n) = n + n\2}
```

concat(0, Vec(x*(1+2*x)/((1-x)*(1-x^2)) + O(x^100))) \\ Altug Alkan, Dec 09 2015

[int(3*n//2) for n in range(101)] # G. C. Greubel, Jun 23 2024

G.f.: x*(1+2*x)/((1-x)*(1-x^2)).
a(-n) = -A007494(n).
a(n) = A049615(n, 2), for n > 2.
From Paul Barry, Sep 04 2003: (Start)
a(n) = (6n - 1 + (-1)^n)/4.
a(n) = floor((3n + 2)/2) - 1 = A001651(n) - 1.
a(n) = sqrt(2) * sqrt( (6n-1) (-1)^n + 18n^2 - 6n + 1 )/4.
a(n) = Sum_{k=0..n} 3/2 - 2*0^k + (-1)^k/2. (End)
a(n) = 3*floor(n/2) + (n mod 2) = A007494(n) - A000035(n). - Reinhard Zumkeller, Apr 04 2005
a(n) = 2 * A004526(n) + A004526(n+1). - Philippe Deléham, Aug 07 2006
a(n) = 1 + ceiling(3*(n-1)/2). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
Row sums of triangle A133083. - Gary W. Adamson, Sep 08 2007
a(n) = (cos(Pi*n) - 1)/4 + 3*n/2. - Bart Snapp (snapp(AT)coastal.edu), Sep 18 2008
A004396(a(n)) = n. - Reinhard Zumkeller, Oct 30 2009
a(n) = floor(n/2) + n. - Gary Detlefs, Mar 19 2010
a(n) = 3n - a(n-1) - 2, for n>0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
a(n) = n + (n-1) - (n-2) + (n-3) - ... 1 = A052928(n) + A008619(n-1). - Jaroslav Krizek, Mar 22 2011
a(n) = a(n-1) + a(n-2) - a(n-3). - Robert G. Wilson v, Mar 28 2011
a(n) = Sum_{k>=0} A030308(n,k) * A003945(k). - Philippe Deléham, Oct 17 2011
a(n) = 2n - ceiling(n/2). - Wesley Ivan Hurt, Oct 25 2013
a(n) = A000217(n) - 2 * A002620(n-1). - Kival Ngaokrajang, Oct 26 2013
a(n) = Sum_{i=1..n} gcd(i, 2). - Wesley Ivan Hurt, Jan 23 2014
a(n) = 2n + floor((-n - (n mod 2))/2). - Wesley Ivan Hurt, Mar 31 2014
A092942(a(n)) = n for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n) = floor(3*n/2). - L. Edson Jeffery, Jan 18 2015
a(n) = A254049(A249745(n)) = (1+A007310(n)) / 2 for n >= 1. - Antti Karttunen, Jan 24 2015
E.g.f.: (3*x*exp(x) - sinh(x))/2. - Ilya Gutkovskiy, Jul 18 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) + log(3)/2. - Amiram Eldar, Dec 04 2021

Better description from N. J. A. Sloane, Aug 01 1998

A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 3, 2, 4, 8, 5, 6, 11, 23, 14, 7, 17, 32, 68, 41, 9, 20, 50, 95, 203, 122, 10, 26, 59, 149, 284, 608, 365, 12, 29, 77, 176, 446, 851, 1823, 1094, 13, 35, 86, 230, 527, 1337, 2552, 5468, 3281, 15, 38, 104, 257, 689, 1580, 4010, 7655, 16403, 9842, 16, 44, 113, 311, 770, 2066, 4739, 12029, 22964, 49208, 29525, 18, 47

Offset: 1

L. Edson Jeffery & Antti Karttunen, Jan 24 2015

This is transposed dispersion of (3n-1), starting from its complement A032766 as the first row of square array A(row,col). Please see the transposed array A191450 for references and background discussion about dispersions.

For any odd number x = A135765(row,col), the result after one combined Collatz step (3x+1)/2 -> x (A165355) is found in this array at A(row+1,col).

			The top left corner of the array:
   1,   3,   4,   6,   7,   9,  10,  12,   13,   15,   16,   18,   19,   21
   2,   8,  11,  17,  20,  26,  29,  35,   38,   44,   47,   53,   56,   62
   5,  23,  32,  50,  59,  77,  86, 104,  113,  131,  140,  158,  167,  185
  14,  68,  95, 149, 176, 230, 257, 311,  338,  392,  419,  473,  500,  554
  41, 203, 284, 446, 527, 689, 770, 932, 1013, 1175, 1256, 1418, 1499, 1661
...

Inverse: A254052.
Transpose: A191450.
Row 1: A032766.
Cf. A007051, A057198, A199109, A199113 (columns 1-4).
Cf. A254046 (row index of n in this array, see also A253786), A253887 (column index).
Array A135765(n,k) = 2*A(n,k) - 1.
Other related arrays: A254055, A254101, A254102.
Cf. also A000079, A000244, A003961, A007310, A032766, A002260, A004736, A038722, A165355, A254050.
Related permutations: A048673, A254053, A183209, A249745, A254103, A254104.

In A(n,k)-formulas below, n is the row, and k the column index, both starting from 1:
A(n,k) = (3 + ( A000244(n) * (2*A032766(k) - 1) )) / 6. - Antti Karttunen after L. Edson Jeffery's direct formula for A191450, Jan 24 2015
A(n,k) = A048673(A254053(n,k)). [Alternative formula.]
A(n,k) = (1/2) * (1 + A003961((2^(n-1)) * A254050(k))). [The above expands to this.]
A(n,k) = (1/2) * (1 + (A000244(n-1) * A007310(k))). [Which further reduces to this, equivalent to L. Edson Jeffery's original formula above.]
A(1,k) = A032766(k) and for n > 1: A(n,k) = (3 * A254051(n-1,k)) - 1. [The definition of transposed dispersion of (3n-1).]
A(n,k) = (1+A135765(n,k))/2, or when expressed one-dimensionally, a(n) = (1+A135765(n))/2.
A(n+1,k) = A165355(A135765(n,k)).
As a composition of related permutations. All sequences interpreted as one-dimensional:
a(n) = A048673(A254053(n)). [Proved above.]
a(n) = A191450(A038722(n)). [Transpose of array A191450.]

A273669 Decimal representation ends with either 2 or 9.

Original entry on oeis.org

2, 9, 12, 19, 22, 29, 32, 39, 42, 49, 52, 59, 62, 69, 72, 79, 82, 89, 92, 99, 102, 109, 112, 119, 122, 129, 132, 139, 142, 149, 152, 159, 162, 169, 172, 179, 182, 189, 192, 199, 202, 209, 212, 219, 222, 229, 232, 239, 242, 249, 252, 259, 262, 269, 272, 279, 282, 289, 292, 299, 302, 309, 312, 319, 322, 329, 332, 339

Offset: 1

Antti Karttunen, Aug 06 2016

Natural numbers not in A273664.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature(1,1,-1).

Sequences A017293 and A017377 interleaved.
Cf. A000035, A001622, A007310, A084967, A126760, A249745, A249746.
Cf. also A273664, A249824, A275716.

Select[Range@ 340, MemberQ[{2, 9}, Mod[#, 10]] &] (* or *)
Table[{10 n + 2, 10 n + 9}, {n, 0, 33}] // Flatten (* or *)
CoefficientList[Series[(-5/(1 - x) + (11 - x)/(-1 + x)^2 - 2/(1 + x))/2, {x, 0, 67}], x] (* Michael De Vlieger, Aug 07 2016 *)

(define (A273669 n) (+ (* 10 (/ (+ (- n 2) (if (odd? n) 1 0)) 2)) (if (odd? n) 2 9)))

a(n) = 10*(((n-2)+A000035(n))/2) + 2 [when n is odd], or + 9 [when n is even].
For n >= 5, a(n) = 2*a(n-2) - a(n-4).
a(n) = A126760(A084967(n)).
a(n) = A249746((3*A249745(n))-1).
Other identities. For all n >= 1:
A084967(n) = 5*A007310(n) = A007310(a(n)).
G.f.: x*(x^2+7*x+2)/((x+1)*(x-1)^2).
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt((1+1/sqrt(5))/2)*phi^2*Pi/10 - log(phi)/(2*sqrt(5)) - log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

A249746 Permutation of natural numbers: a(n) = A126760(A249735(n)) = A249824(A064216(n)).

Original entry on oeis.org

1, 2, 3, 4, 9, 5, 6, 12, 7, 8, 19, 10, 17, 42, 11, 13, 22, 26, 14, 29, 15, 16, 59, 18, 41, 32, 20, 31, 39, 21, 23, 92, 40, 24, 49, 25, 27, 82, 48, 28, 209, 30, 45, 52, 33, 63, 62, 54, 34, 109, 35, 36, 129, 37, 38, 69, 43, 68, 142, 70, 57, 72, 115, 44, 79, 46, 85, 292, 47, 50, 89, 74, 73, 202, 51, 53, 159, 87, 55, 99, 107, 56, 152, 58, 97, 192, 60

Offset: 1

Antti Karttunen, Nov 23 2014

nonn

Permutation obtained from the odd bisection of A003961 (or from the odd bisection of A048673).

			a(5) = 9 because of the following. 2*A064216(5) = 2*4 = 8 = 2^3. We replace the prime factor 2 of 8 with the next prime 3 to get 3^3, then replace 3 with 5 to get 5^3 = 125. The smallest prime factor of 125 is 5. 125 is the 9th term of A084967: 5, 25, 35, 55, 65, 85, 95, 115, 125, ..., thus a(5) = 9.

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

Inverse: A249745.
Row 2 of A251722.
Cf. A003961, A007310, A048673, A084967, A126760, A249735, A064216, A249824.
Cf. also A273664, A273669.

t = PositionIndex[FactorInteger[#][[1, 1]] & /@ Range[10^6]]; f[n_] := Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Flatten@ Map[Position[Lookup[t, FactorInteger[#][[1, 1]] ], #] &[f@ f[2 #]] &, Table[Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1], {n, 87}]] (* Michael De Vlieger, Jul 25 2016, Version 10 *)

(define (A249746 n) (define (Ainv_of_A007310off0 n) (+ (* 2 (floor->exact (/ n 6))) (/ (- (modulo n 6) 1) 4))) (+ 1 (Ainv_of_A007310off0 (A003961 (+ n n -1)))))

a(n) = 1 + f(A003961(2n - 1)), where f(n) = 2*floor[n/6] + ((n mod 6)-1)/4. [Here 1 + f(A007310(n)) = n.]
a(n) = A126760(A249735(n)). - Antti Karttunen, Jul 25 2016
As a composition of related permutations:
a(n) = A249824(A064216(n)).
Other identities. For all n >= 1:
A249735(n) = A007310(a(n)).
a(3n-1) = A273669(a(n)) and a(A254049(n)) = A273664(a(n)). - Antti Karttunen, Aug 07 2016

cp's OEIS Frontend

A254053 Square array: A(row,col) = 2^(row-1) * ((2*A249745(col))-1) = A064216(A254051(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A254050 Permutation of odd numbers: a(n) = (2*(A249745(n))) - 1 = A064989(A007310(n)).

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A254118 Permutation of natural numbers: a(n) = A249745(1+A254103(n)) - 1.

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A064989 Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

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A048673 Permutation of natural numbers: a(n) = (A003961(n)+1) / 2 [where A003961(n) shifts the prime factorization of n one step towards larger primes].

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A003602 Kimberling's paraphrases: if n = (2k-1)*2^m then a(n) = k.

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A254051 Square array A by downward antidiagonals: A(n,k) = (3 + 3^n(2floor(3*k/2) - 1))/6, n,k >= 1; read as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...