A249746 -id:A249746 - cp's OEIS frontend

A273664 a(n) = A249746(A032766(n)).

1, 3, 4, 5, 6, 7, 8, 10, 17, 11, 13, 26, 14, 15, 16, 18, 41, 20, 31, 21, 23, 40, 24, 25, 27, 48, 28, 30, 45, 33, 63, 54, 34, 35, 36, 37, 38, 43, 68, 70, 57, 115, 44, 46, 85, 47, 50, 74, 73, 51, 53, 87, 55, 107, 56, 58, 97, 60, 180, 61, 64, 96, 83, 65, 66, 67, 71, 114, 101, 100, 75, 110, 136, 108, 76, 77, 78, 80, 124, 81

Offset: 1

Antti Karttunen, Aug 06 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A032766, A249745, A249746, A249824, A254049, A254050, A275716.

Cf. also A273669 (natural numbers not in this sequence).

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 #]] &, Map[Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@#, Last@#] &@ Transpose@ FactorInteger[2 # - 1] &, Floor[#/2] + # & /@ Range@ 80]] (* Michael De Vlieger, Aug 07 2016, Version 10 *)

(define (A273664 n) (A249746 (A032766 n)))

a(n) = A249746(A032766(n)).
a(n) = A249824(A254050(n)).
a(n) = A249746(A254049(A249745(n))).

A254117 Permutation of natural numbers: a(n) = A254104(A249746(1+n)-1).

Original entry on oeis.org

1, 2, 3, 6, 5, 4, 10, 7, 11, 21, 9, 31, 16, 15, 23, 27, 63, 13, 43, 8, 19, 53, 14, 37, 45, 47, 95, 26, 22, 17, 511, 127, 12, 91, 39, 18, 85, 62, 29, 2047, 30, 38, 111, 20, 54, 75, 42, 55, 157, 35, 46, 174, 25, 79, 24, 59, 77, 303, 123, 94, 78, 167, 87, 223, 61, 103, 699, 191, 41, 107, 183, 125, 431, 28, 71, 84, 60, 93, 110, 767, 51

Offset: 1

Antti Karttunen, Feb 05 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A254118.

Other related permutations: A249746, A254104, A254115.

from sympy import factorint, nextprime
from operator import mul
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))/2
def a254104(n):
    if n==0: return 0
    if n%3==0: return 1 + 2*a254104(2*n/3 - 1)
    elif n%3==1: return 1 + 2*a254104(2*(n - 1)/3)
    else: return 2*a254104((n - 2)/3 + 1)
def a254115(n): return a254104(a048673(n))
def a(n): return (a254115(2*n + 1) - 1)/2 # Indranil Ghosh, Jun 06 2017

a(n) = A254104(A249746(1+n)-1).

a(n) = (A254115((2*n)+1)-1) / 2. [Obtained also from the odd bisection of A254115.]

A126760 a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 5, 3, 2, 1, 6, 1, 7, 2, 3, 4, 8, 1, 9, 5, 1, 3, 10, 2, 11, 1, 4, 6, 12, 1, 13, 7, 5, 2, 14, 3, 15, 4, 2, 8, 16, 1, 17, 9, 6, 5, 18, 1, 19, 3, 7, 10, 20, 2, 21, 11, 3, 1, 22, 4, 23, 6, 8, 12, 24, 1, 25, 13, 9, 7, 26, 5, 27, 2, 1, 14, 28, 3, 29, 15, 10, 4, 30, 2

Offset: 0

N. J. A. Sloane, Feb 19 2007

nonn

For further information see A126759, which provided the original motivation for this sequence.
From Antti Karttunen, Jan 28 2015: (Start)
The odd bisection of the sequence gives A253887, and the even bisection gives the sequence itself.
A254048 gives the sequence obtained when this sequence is restricted to A007494 (numbers congruent to 0 or 2 mod 3).
For all odd numbers k present in square array A135765, a(k) = the column index of k in that array. (End)
A322026 and this sequence (without the initial zero) are ordinal transforms of each other. - Antti Karttunen, Feb 09 2019
Also ordinal transform of A065331 (after the initial 0). - Antti Karttunen, Sep 08 2024

Antti Karttunen, Table of n, a(n) for n = 0..19683

One less than A126759.
Cf. A003586, A003602, A007310, A064989, A249746, A253887, A254048, A273669, A322317, A323881 (Dirichlet inverse), A323882.
Cf. A347233 (Möbius transform) and also A349390, A349393, A349395 for other Dirichlet convolutions.
Ordinal transform of A065331 and of A322026 (after the initial 0).
Related arrays: A135765, A254102.

f[n_] := Block[{a}, a[0] = 0; a[1] = a[2] = a[3] = 1; a[x_] := Which[EvenQ@ x, a[x/2], Mod[x, 3] == 0, a[x/3], Mod[x, 6] == 1, 2 (x - 1)/6 + 1, Mod[x, 6] == 5, 2 (x - 5)/6 + 2]; Table[a@ i, {i, 0, n}]] (* Michael De Vlieger, Feb 03 2015 *)

A126760(n)={n&&n\=3^valuation(n,3)<M. F. Hasler, Jan 19 2016

a(n) = A126759(n)-1. [The original definition.]
From Antti Karttunen, Jan 28 2015: (Start)
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.
Or with the last clause represented in another way:
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n-1) = 2n.
Other identities. For all n >= 1:
a(n) = A253887(A003602(n)).
a(6n-3) = a(4n-2) = a(2n-1) = A253887(n).
(End)
a(n) = A249746(A003602(A064989(n))). - Antti Karttunen, Feb 04 2015
a(n) = A323882(4*n). - Antti Karttunen, Apr 18 2022

Name replaced with an independent recurrence and the old description moved to the Formula section - Antti Karttunen, Jan 28 2015

A135764 Distribute the natural numbers in columns based on the occurrence of "2" in each prime factorization; square array A(row,col) = 2^(row-1) * ((2*col)-1), read by descending antidiagonals.

Original entry on oeis.org

1, 3, 2, 5, 6, 4, 7, 10, 12, 8, 9, 14, 20, 24, 16, 11, 18, 28, 40, 48, 32, 13, 22, 36, 56, 80, 96, 64, 15, 26, 44, 72, 112, 160, 192, 128, 17, 30, 52, 88, 144, 224, 320, 384, 256, 19, 34, 60, 104, 176, 288, 448, 640, 768, 512, 21, 38, 68, 120, 208, 352, 576, 896, 1280, 1536, 1024, 23, 42, 76, 136, 240, 416, 704, 1152, 1792, 2560, 3072, 2048, 25, 46, 84, 152, 272, 480, 832, 1408, 2304, 3584, 5120, 6144, 4096, 27, 50, 92, 168, 304, 544, 960, 1664, 2816

Offset: 1

Alford Arnold, Nov 29 2007

The array in A135764 is identical to the array in A054582 [up to the transposition and different indexing. - Clark Kimberling, Dec 03 2010; comment amended by Antti Karttunen, Feb 03 2015; please see the illustration in Example section].
The array gives a bijection between the natural numbers N and N^2. A more usual bijection is to take the natural numbers A000027 and write them in the usual OEIS square array format. However this bijection has the advantage that it can be formed by iterating the usual bijection between N and 2N. - Joshua Zucker, Nov 04 2011
The array can be used to determine the configurations of k-th Towers of Hanoi moves, by labeling odd row terms C,B,A,C,B,A,... and even row terms B,C,A,B,C,A,.... Then given k equal to or greater than term "a" in each n-th row, but less than the next row term, record the label A, B, or C for term "a". This denotes the peg position for the disc corresponding to the n-th row. For example, with k = 25, five discs are in motion since the binary for 25 = 11001, five bits. We find that 25 in row 5 is greater than 16 labeled C, but less than 48. Thus, disc 5 is on peg C. In the 4th row, 25 is greater than 24 (a C), but less than 40, so goes onto the C peg. Similarly, disc 3 is on A, 2 is on A, and disc 1 is on A. Thus, discs 2 and 3 are on peg A, while 1, 4, and 5 are on peg C. - Gary W. Adamson, Jun 22 2012
Shares with arrays A253551 and A254053 the property that A001511(n) = k for all terms n on row k and when going downward in each column, terms grow by doubling. - Antti Karttunen, Feb 03 2015
Let P be the infinite palindromic word having initial word 0 and midword sequence (1,2,3,4,...) = A000027. Row n of the array A135764 gives the positions of n-1 in S. ("Infinite palindromic word" is defined at A260390.) - Clark Kimberling, Aug 13 2015
The probability distribution series 1 = 2/3 + 4/15 + 16/255 + 256/65535 + ... + A001146(n-1)/A051179(n) governs the proportions of terms in A001511 from row n of the array. In A001511(1..15) there are ((2/3) * 15) = ten terms from row one of the array, ((4/15) * 15) = four terms from row two, and ((16/255) * 15) = one (rounded), giving one term from row three (a 4). - Gary W. Adamson, Dec 16 2021
From Gary W. Adamson, Dec 30 2021: (Start)
Subarrays representing the number of divisors of an integer can be mapped on the table. For 60, write the odd divisors on the top row: 1, 3, 5, 15. Since 60 has 12 divisors, let the left column equal 1, 2, 4, where 4 is the highest power of 2 dividing 60. Multiplying top row terms by left column terms, we get the result:
  1   3   5  15
  2   6  10  30
  4  12  20  60. (End)

			The table begins
   1,  3,   5,   7,   9,  11,  13,  15,  17,  19,  21,  23, ...
   2,  6,  10,  14,  18,  22,  26,  30,  34,  38,  42,  46, ...
   4, 12,  20,  28,  36,  44,  52,  60,  68,  76,  84,  92, ...
   8, 24,  40,  56,  72,  88, 104, 120, 136, 152, 168, 184, ...
  16, 48,  80, 112, 144, 176, 208, 240, 272, 304, 336, 368, ...
  32, 96, 160, 224, 288, 352, 416, 480, 544, 608, 672, 736, ...
etc.
For n = 6, we have [A002260(6), A004736(6)] = [3, 1] (i.e., 6 corresponds to location 3,1 (row,col) in above table) and A(3,1) = A000079(3-1) * A005408(1-1) = 2^2 * 1 = 4.
For n = 13, we have [A002260(13), A004736(13)] = [3, 3] (13 corresponds to location 3,3 (row,col) in above table) and A(3,3) = A000079(3-1) * A005408(3-1) = 2^2 * 5 = 20.
For n = 23, we have [A002260(23), A004736(23)] = [2, 6] (23 corresponds to location 2,6) and A(2,6) = A000079(2-1) * A005408(6-1) = 2^1 * 11 = 22.

Transpose: A054582.
Inverse permutation: A249725.
Column 1: A000079.
Row 1: A005408.
Cf. A001511 (row index), A003602 (column index, both one-based).
Related arrays: A135765, A253551, A254053, A254055.
Cf. A000027, A002260, A004736, A064989, A135766, A249746.
Cf. also permutations A246675, A246676, A249741, A249811, A249812.
Cf. A260390.
Cf. A001146, A051179.

seq(seq(2^(j-1)*(2*(i-j)+1),j=1..i),i=1..20); # Robert Israel, Feb 03 2015

f[n_] := Block[{i, j}, {1}~Join~Flatten@ Last@ Reap@ For[j = 1, j <= n, For[i = j, i > 0, Sow[2^(j - i - 1)*(2 i + 1)], i--], j++]]; f@ 10 (* Michael De Vlieger, Feb 03 2015 *)

a(n) = {s = ceil((1 + sqrt(1 + 8*n)) / 2); r = n - binomial(s-1, 2) - 1;k = s - r - 2; 2^r * (2 * k + 1) } \\ David A. Corneth, Feb 05 2015

(define (A135764 n) (A135764bi (A002260 n) (A004736 n)))
(define (A135764bi row col) (* (A000079 (- row 1)) (+ -1 col col)))
;; Antti Karttunen, Feb 03 2015

From Antti Karttunen, Feb 03 2015: (Start)
A(row, col) = 2^(row-1) * ((2*col)-1) = A000079(row-1) * A005408(col-1).
A(row,col) = A064989(A135765(row,A249746(col))).
A(row,col) = A(row+1,col)/2 [discarding the topmost row and halving the rest of terms gives the array back].
A(row,col) = A(row,col+1) - A000079(row) [discarding the leftmost column and subtracting 2^{row number} from the rest of terms gives the array back].
(End)
G.f.: ((2*x+1)*Sum_{i>=0} 2^i*x^(i*(i+1)/2) + 2*(1-2*x)*Sum_{i>=0} i*x^(i*(i+1)/2) + (1-6*x)*Sum_{i>=0} x^(i*(i+1)/2) - 1 - 2*x)*x/(1-2*x)^2. These sums are related to Jacobi theta functions. - Robert Israel, Feb 03 2015

More terms from Sean A. Irvine, Nov 23 2010

Name amended and the illustration of array in the example section transposed by Antti Karttunen, Feb 03 2015

A254049 Odd bisection of A048673: a(n) = A048673(2*n-1).

Original entry on oeis.org

1, 3, 4, 6, 13, 7, 9, 18, 10, 12, 28, 15, 25, 63, 16, 19, 33, 39, 21, 43, 22, 24, 88, 27, 61, 48, 30, 46, 58, 31, 34, 138, 60, 36, 73, 37, 40, 123, 72, 42, 313, 45, 67, 78, 49, 94, 93, 81, 51, 163, 52, 54, 193, 55, 57, 103, 64, 102, 213, 105, 85, 108, 172, 66, 118, 69, 127, 438, 70, 75, 133, 111, 109, 303

Offset: 1

Antti Karttunen, Jan 24 2015

nonn

Shift the prime factorization of odd numbers one step towards larger primes, add one and divide by two.

			For n = 8, the eighth odd number is 2*8 - 1 = 15 = 3*5 = prime(2) * prime(3). By adding one to both prime indices, we get prime(3) * prime(4) = 5*7 = 35, and (35+1)/2 = 18, thus a(8) = 18. Here prime(n) = A000040(n).

Cf. A032766 (omitting the initial 0, the same sequence sorted into ascending order).
Also a permutation of A253888.
Cf. A000040, A003961, A048673, A249735, A249746, A254050, A254051, A254053.

a(n) = A048673(2*n-1) = (1+A003961(2*n-1)) / 2 = (1+A249735(n)) / 2.

a(n) = A032766(A249746(n)).

A249745 Permutation of natural numbers: a(n) = (1 + A064989(A007310(n))) / 2.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 5, 12, 15, 8, 16, 19, 21, 22, 13, 24, 11, 27, 30, 17, 31, 34, 36, 18, 37, 40, 20, 42, 28, 26, 45, 49, 51, 52, 54, 55, 29, 33, 25, 14, 57, 64, 43, 66, 69, 39, 35, 70, 75, 44, 76, 48, 79, 82, 61, 84, 23, 87, 90, 47, 46, 91, 96, 97, 99, 58, 56, 60, 100, 62, 73, 72, 106, 112, 114, 115, 65, 117, 120, 38, 94, 121

Offset: 1

Antti Karttunen, Nov 23 2014

nonn

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

Inverse: A249746.
Row 2 of A251721.
Cf. A007310, A048673, A064989, A249823, A249735.

a249745[n_Integer] := Module[{f, p, a064989, a007310, a},
  f[x_] := Transpose@FactorInteger[x];
  p[x_] := Which[
    x == 1, 1,
    x == 2, 1,
    True, NextPrime[x, -1]];
  a064989[x_] := Times @@ Power[p /@ First[f[x]], Last[f[x]]];
  a007310[x_] := Select[Range[x], MemberQ[{1, 5}, Mod[#, 6]] &];
  a[x_] := (1 + a064989 /@ a007310[x])/2;
a[n]]; a249745[252] (* Michael De Vlieger, Dec 18 2014, after Harvey P. Dale at A007310 *)

A249745(n)=A064989(A007310(n))\2+1 \\ M. F. Hasler, Jan 19 2016

(define (A249745 n) (/ (+ 1 (A064989 (A007310 n))) 2))

a(n) = (A064989(A007310(n)) + 1) / 2.
a(n) = A048673(A249823(n)), as a composition of related permutations.
A007310(n) = A249735(a(n)) for all n >= 1. (This is the permutation which sorts the terms of A249735 into an ascending order, as they occur in A007310.)

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

A251722 Square array of permutations: A(row,col) = A249822(row+1, A249821(row, col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 3, 2, 1, 4, 4, 3, 2, 1, 8, 9, 4, 3, 2, 1, 6, 5, 5, 4, 3, 2, 1, 14, 6, 6, 5, 4, 3, 2, 1, 13, 12, 7, 6, 5, 4, 3, 2, 1, 11, 7, 8, 7, 6, 5, 4, 3, 2, 1, 7, 8, 14, 8, 7, 6, 5, 4, 3, 2, 1, 23, 19, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 10, 10, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 17, 21, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 42, 11, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Antti Karttunen, Dec 07 2014

These are the "first differences" between permutations of array A249822, in a sense that by composing the first k rows of this array [from right to left, as in a(n) = row_k(...(row_2(row_1(n))))], one obtains row k+1 of A249822.

On row n the first non-fixed term is A250474(n+1) at position A250474(n), i.e., on row 1 it is 5 at n=4, on row 2 it is 9 at n=5, on row 3 it is 14 at n=9, etc. All the previous A250473(n) terms are fixed.

			The top left corner of the array:
1, 2, 3, 5, 4, 8, 6, 14, 13, 11, 7, 23, 9, 17, 18, 41, 10, 38, 12, 32, ...
1, 2, 3, 4, 9, 5, 6, 12, 7, 8, 19, 10, 17, 42, 11, 13, 22, 26, 14, 29, ...
1, 2, 3, 4, 5, 6, 7, 8, 14, 9, 10, 21, 11, 12, 13, 15, 33, 16, 25, 17, ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 28, 14, 15, 16, 17, 18, 19, ...
...

Inverse permutations can be found from array A251721.
Row 1: A048673, Row 2: A249746, Row 3: A250476.
Cf. A000027, A002260, A004736, A078898, A083221, A246277, A246278, A249821, A249822, A250473, A250474.

(define (A251722bi row col) (A249822bi (+ row 1) (A249821bi row col)))
(define (A251722 n) (A251722bi (A002260 n) (A004736 n)))
;; Code for A249821bi and A249822bi given in A249821, A249822.

A(row,col) = A249822(row+1, A249821(row, col)).

A(row,col) = A078898(A246278(row+1, A246277(A083221(row, col)))).

A253887 Row index of n in A191450: a(3n) = 2n, a(3n+1) = 2n+1, a(3n+2) = a(n+1).

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 5, 2, 6, 7, 3, 8, 9, 1, 10, 11, 4, 12, 13, 5, 14, 15, 2, 16, 17, 6, 18, 19, 7, 20, 21, 3, 22, 23, 8, 24, 25, 9, 26, 27, 1, 28, 29, 10, 30, 31, 11, 32, 33, 4, 34, 35, 12, 36, 37, 13, 38, 39, 5, 40, 41, 14, 42, 43, 15, 44, 45, 2, 46, 47, 16, 48, 49, 17, 50, 51, 6, 52, 53, 18, 54, 55, 19, 56, 57, 7

Offset: 1

Antti Karttunen, Jan 22 2015

nonn

a(n) gives the row index of n in square array A191450, or equally, the column index of n in A254051.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Odd bisection of A126760.
Cf. A254046 (the corresponding column index).
Cf. A003602, A032766, A064216, A253786, A253893, A249746.
Cf. A191450, A254051, A254047, A254052, A254104.

def a(n):
    if n%3==0: return 2*n//3
    elif n%3==1: return 2*(n - 1)//3 + 1
    else: return a((n - 2)//3 + 1)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

a(3n) = 2n, a(3n+1) = 2n+1, a(3n+2) = a(n+1).
a(n) = A126760(2n-1).
a(n) = A249746(A003602(A064216(n))). - Antti Karttunen, Feb 04 2015

A249824 Permutation of natural numbers: a(n) = A078898(A003961(A003961(2*n))).

Original entry on oeis.org

1, 2, 3, 9, 4, 12, 5, 42, 17, 19, 6, 59, 7, 22, 26, 209, 8, 82, 10, 92, 31, 29, 11, 292, 41, 32, 115, 109, 13, 129, 14, 1042, 40, 39, 48, 409, 15, 49, 45, 459, 16, 152, 18, 142, 180, 52, 20, 1459, 57, 202, 54, 159, 21, 572, 63, 542, 68, 62, 23, 642, 24, 69, 213

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

			a(4) = 9 because of the following. 2n = 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(4) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..255
Index entries for sequences that are permutations of the natural numbers

Inverse: A249823.
Row 3 of A249822.
Cf. A003961, A048673, A078898, A084967, A243071, A246278, A249734, A249746, A249826, A250475, A275716.
Cf. also A273664, A273669.

t = PositionIndex[FactorInteger[#][[1, 1]] & /@ Range[10^4]]; f[n_] := Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Flatten@ Table[Position[Lookup[t, FactorInteger[#][[1, 1]] ], #] &[f@ f[2 n]], {n, 120}] (* Michael De Vlieger, Jul 25 2016, Version 10 *)

(define (A249824 n) (A078898 (A003961 (A003961 (* 2 n)))))

a(n) = A078898(A003961(A003961(2*n))) = A078898(A003961(A249734(n))).
a(n) = A078898(A246278(3,n)).
As a composition of other permutations:
a(n) = A249746(A048673(n)).
a(n) = A250475(A249826(n)).
a(n) = A275716(A243071(n)).
Other identities. For all n >= 1:
a(2n) = A273669(a(n)) and a(A003961(n)) = A273664(a(n)). -- Antti Karttunen, Aug 07 2016

cp's OEIS Frontend

A273664 a(n) = A249746(A032766(n)).

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A254117 Permutation of natural numbers: a(n) = A254104(A249746(1+n)-1).

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A126760 a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.

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A135764 Distribute the natural numbers in columns based on the occurrence of "2" in each prime factorization; square array A(row,col) = 2^(row-1) * ((2*col)-1), read by descending antidiagonals.

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Maple

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A254049 Odd bisection of A048673: a(n) = A048673(2*n-1).

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A249745 Permutation of natural numbers: a(n) = (1 + A064989(A007310(n))) / 2.

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A273669 Decimal representation ends with either 2 or 9.

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A251722 Square array of permutations: A(row,col) = A249822(row+1, A249821(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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