A245608 -id:A245608 - cp's OEIS frontend

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].

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

A245612 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = 3*a(n)-1, a(2n+1) = A254049(a(n)); composition of A048673 and A163511.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jul 28 2014

Note the indexing: the domain starts from 0, while the range excludes zero.
From Antti Karttunen, Jul 25 2016: (Start)
This sequence can be represented as a binary tree. Each left hand child is obtained by applying A016789(n-1) when the parent contains n (i.e., multiply by 3, subtract one), and each right hand child is obtained by applying A254049 to the parent's contents:
                                     1
                                     |
                  ...................2...................
                 5                                       3
      14......../ \........13                  8......../ \........4
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  41       63         38       25         23       18         11       6
122 313 188  172   113  123  74  61     68  88   53  39     32  28   17 7
etc.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A245611.

Cf. A016789, A048673, A163511, A243065-A243066, A243505-A243506, A245608, A245610, A244153, A244154, A254049.

Table[(Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ If[n == 0, 1, Prime[#] Product[Prime[m]^(Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]][[-m]]), {m, #}] &[DigitCount[n, 2, 1]]], {n, 0, 63}] (* Michael De Vlieger, Jul 25 2016 *)

(define (A245612 n) (A048673 (A163511 n))) ;; offset 0, a(0) = 1.

a(n) = A048673(A163511(n)).

a(0) = 1, a(1) = 2, a(2n) = 3*a(n)-1, a(2n+1) = A254049(a(n)). - Antti Karttunen, Jul 25 2016

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.

Original entry on oeis.org

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)

A245606 Permutation of natural numbers: a(1) = 1, a(2n) = 1 + A003961(a(n)), a(2n+1) = A003961(1+a(n)). [Where A003961(n) shifts the prime factorization of n one step left].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 7, 8, 15, 16, 11, 26, 21, 22, 13, 12, 27, 28, 25, 36, 81, 82, 19, 14, 45, 52, 125, 56, 39, 40, 29, 18, 33, 46, 17, 126, 99, 100, 31, 50, 51, 226, 41, 626, 129, 130, 89, 24, 63, 34, 35, 176, 87, 154, 59, 344, 825, 298, 115, 86, 189, 190, 43, 32, 105, 76, 23, 66, 57, 88, 53, 20

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

The even bisection halved gives A245608. The odd bisection incremented by one and halved gives A245708.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245605.
Cf. A003961, A243501, A064216, A245608, A245706, A245708.
Similar entanglement permutations: A244319, A156552, A243071, A243288, A243344, A243346, A244322, A245604, A245614, A227413, A237126.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A245606(n) = if(1==n,1,if(0==(n%2),1+A003961(A245606(n/2)),A003961(1+A245606((n-1)/2))))

;; With memoization-macro definec.
(definec (A245606 n) (cond ((= 1 n) 1) ((even? n) (A243501 (A245606 (/ n 2)))) (else (A003961 (+ 1 (A245606 (/ (- n 1) 2)))))))

a(1) = 1, a(2n) = A243501(a(n)), a(2n+1) = A003961(1+a(n)).
As a composition of related permutations:
a(n) = A064216(A245608(n)).

A245607 Permutation of natural numbers, the even bisection of A245605 halved: a(n) = A245605(2*n)/2.

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 13, 6, 17, 37, 8, 25, 7, 10, 69, 33, 26, 11, 41, 16, 277, 45, 18, 65, 21, 14, 1109, 15, 52, 73, 57, 74, 35, 209, 82, 293, 141, 34, 53, 329, 12, 1173, 31, 36, 213, 149, 104, 43, 49, 20, 145, 173, 138, 81, 581, 114, 553, 71, 90, 133, 101, 282, 19, 325, 24, 457, 165, 50, 77, 97, 62, 105, 555, 42

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245608.

Cf. A064216, A064989, A245705, A245707, A245605, A245609-A245610, A245611, A244153, A243065-A243066.

A064989(n) = my(f = factor(n)); for(i=1, #f~, if((2 == f[i,1]),f[i,1] = 1,f[i,1] = precprime(f[i,1]-1))); factorback(f);
A245605(n) = if(1==n, 1, if(0==(n%2), 2*A245605(A064989(n-1)), 1+(2*A245605(A064989(n)-1))));
A245607(n) = A245605(2*n)/2;
for(n=1, 10001, write("b245607.txt", n, " ", A245607(n)));

(define (A245607 n) (A245605 (A064216 n)))

a(n) = A245605(2*n)/2.
As a composition of related permutations:
a(n) = A245605(A064216(n)).
a(n) = A245705(A245707(n)).

A243501 Permutation of even numbers: a(n) = 2*A048673(n).

Original entry on oeis.org

2, 4, 6, 10, 8, 16, 12, 28, 26, 22, 14, 46, 18, 34, 36, 82, 20, 76, 24, 64, 56, 40, 30, 136, 50, 52, 126, 100, 32, 106, 38, 244, 66, 58, 78, 226, 42, 70, 86, 190, 44, 166, 48, 118, 176, 88, 54, 406, 122, 148, 96, 154, 60, 376, 92, 298, 116, 94, 62, 316, 68, 112, 276, 730, 120

Offset: 1

Antti Karttunen, Jul 21 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A243502, A245447, A245448, A244319, A245605-A245606, A245607-A245608.

a(n) = 2*A048673(n).
a(n) = A003961(n) + 1.
a(n) = A243502(A245447(n)).

A245610 Permutation of natural numbers: a(n) = A048673(A244319(n)).

Original entry on oeis.org

1, 3, 2, 13, 8, 4, 26, 7, 5, 28, 14, 172, 149, 25, 41, 18, 635, 102, 113, 1194, 11, 43, 428, 22, 17, 6, 77, 88, 71, 259, 527, 130, 227, 48, 74, 12, 677, 235, 20, 688, 68, 634, 5711, 61, 50, 1593, 1490, 27612, 59, 39, 29, 63, 11438, 10119, 4748, 9, 344, 238, 413, 1602, 941, 69

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245609.

Cf. A003961, A048673, A244319, A245607-A245608.

A048673(n) = (1+A003961(n))/2;
A245610(n) = A048673(A244319(n)); \\ For the other needed functions, see under A244319.
for(n=1, 10001, write("b245610.txt", n, " ", A245610(n)));

(define (A245610 n) (A048673 (A244319 n)))

a(n) = A048673(A244319(n)).

A245609 Permutation of natural numbers: a(n) = A244319(A064216(n)).

Original entry on oeis.org

1, 3, 2, 6, 9, 26, 8, 5, 56, 344, 21, 36, 4, 11, 204, 86, 25, 16, 176, 39, 518, 24, 125, 1376, 14, 7, 1268, 10, 51, 3186, 126, 1015, 298, 476, 305, 3204, 590, 115, 50, 5636, 15, 7118, 22, 825, 162, 2388, 153, 34, 626, 45, 4356, 144, 301, 156, 4374, 131, 816, 454, 49, 260, 44, 995, 52, 168, 81

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245610.

Cf. A064216, A064989, A244319, A245607-A245608.

A064216(n) = A064989((2*n)-1);
A245609(n) = A244319(A064216(n)); \\ For the other needed functions, see A244319.
for(n=1, 10001, write("b245609.txt", n, " ", A245609(n)));

(define (A245609 n) (A244319 (A064216 n)))

a(n) = A244319(A064216(n)).

A245708 Permutation of natural numbers, the odd bisection of A245606 incremented by one and halved: a(n) = (1+A245606((2*n)-1))/2.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 6, 11, 7, 14, 13, 41, 10, 23, 63, 20, 15, 17, 9, 50, 16, 26, 21, 65, 45, 32, 18, 44, 30, 413, 58, 95, 22, 53, 12, 29, 27, 38, 66, 221, 52, 122, 48, 77, 115, 83, 748, 179, 69, 263, 25, 365, 39, 113, 153, 176, 130, 158, 508, 1007, 247, 140, 78, 242, 97, 59, 33, 89, 72, 68, 36, 47, 49, 188, 28

Offset: 1

Antti Karttunen, Jul 30 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245707.

Cf. A003961, A048673, A245606, A245608, A245705, A245709.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A245606(n) = if(1==n, 1, if(0==(n%2), 1+A003961(A245606(n/2)), A003961(1+A245606((n-1)/2))));
A245708(n) = (1+A245606((2*n)-1))/2;
for(n=1, 10001, write("b245708.txt", n, " ", A245708(n)))

(define (A245708 n) (* (/ 1 2) (+ 1 (A245606 (-1+ (* 2 n))))))

(define (A245708 n) (if (= 1 n) n (A048673 (+ 1 (A245606 (- n 1))))))

a(n) = (1+A245606((2*n)-1))/2.
As a composition of related permutations:
a(1) = 1, and for n > 1, a(n) = A048673(1+A245606(n-1)).
a(n) = A245608(A245705(n)).
Other identities:
For all n >= 0, a(2^n) = A245608(2^n). Moreover, A245709 gives all such k that a(k) = A245608(k).

cp's OEIS Frontend

A245706 Permutation of natural numbers: a(n) = A245707(A245608(n)).

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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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A245612 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = 3*a(n)-1, a(2n+1) = A254049(a(n)); composition of A048673 and A163511.

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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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A245606 Permutation of natural numbers: a(1) = 1, a(2n) = 1 + A003961(a(n)), a(2n+1) = A003961(1+a(n)). [Where A003961(n) shifts the prime factorization of n one step left].

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A245607 Permutation of natural numbers, the even bisection of A245605 halved: a(n) = A245605(2*n)/2.

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A243501 Permutation of even numbers: a(n) = 2*A048673(n).

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Formula

A245610 Permutation of natural numbers: a(n) = A048673(A244319(n)).