A243062 -id:A243062 - 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

A243065 Permutation of natural numbers, the odd bisection of A241909 halved; equally, a composition of A064216 and A241909: a(n) = A241909(A064216(n)).

Original entry on oeis.org

1, 2, 4, 8, 3, 16, 32, 9, 64, 128, 27, 256, 6, 5, 512, 1024, 81, 18, 2048, 243, 4096, 8192, 25, 16384, 12, 729, 32768, 54, 2187, 65536, 131072, 125, 162, 262144, 6561, 524288, 1048576, 15, 36, 2097152, 7, 4194304, 486, 19683, 8388608, 108, 59049, 1458, 16777216, 625, 33554432, 67108864, 75

Offset: 1

Antti Karttunen, Jun 01 2014

nonn

Are there any other fixed points than 1, 2, 18 and 72?

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Inverse: A243066.

Cf. A064216, A241909, A243505-A243506, A244152-A244154, A243061-A243062, A006254.

(define (A243065 n) (A241909 (A064216 n)))

a(1) = 1, and for n>=2, a(n) = A241909(2n-1)/2. Equally, a(n) = ceiling(A241909(2n-1)/2) for all n.
As a composition of related permutations:
a(n) = A241909(A064216(n)).
a(n) = A241909(A243061(A241909(n))).
For all n, a(A006254(n)) = 2^n.

A243066 Permutation of natural numbers, the even bisection of A241909 incremented by one and halved; equally, a composition of A241909 and A048673: a(n) = A048673(A241909(n)).

Original entry on oeis.org

1, 2, 5, 3, 14, 13, 41, 4, 8, 63, 122, 25, 365, 313, 38, 6, 1094, 18, 3281, 172, 188, 1563, 9842, 61, 23, 7813, 11, 1201, 29525, 123, 88574, 7, 938, 39063, 113, 39, 265721, 195313, 4688, 666, 797162, 858, 2391485, 8404, 74, 976563, 7174454, 85, 68, 88, 23438, 58825, 21523361, 28

Offset: 1

Antti Karttunen, Jun 01 2014

nonn

For n > 1, 2n is found in A241909 from the position (2*a(n))-1. I.e., A241909((2*a(n))-1) = 2n for all n >= 2.
Or in other words, a(n) gives the position in the odd bisection of A241909 where 2n is located at.
Are there any other fixed points than 1, 2, 18 and 72?

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Inverse: A243065.

Cf. A048673, A241909, A243505-A243506, A244152-A244154, A243061-A243062.

a(1) = 1, a(n) = (A241909(2*n)+1)/2.
As a composition of related permutations:
a(n) = A048673(A241909(n)).
a(n) = A241909(A243062(A241909(n))).
For all n>=1, a(2^n) = A006254(n).

A246261 Numbers n such that A003961(n) is of the form 4k+1.

Original entry on oeis.org

1, 3, 4, 9, 10, 11, 12, 13, 14, 16, 23, 25, 27, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 42, 44, 47, 48, 49, 52, 56, 58, 59, 64, 69, 71, 75, 81, 82, 83, 85, 86, 89, 90, 92, 93, 95, 97, 99, 100, 102, 105, 106, 107, 108, 109, 110, 111, 114, 117, 119, 120, 121, 122, 124, 126, 130, 131, 132, 133, 134, 136, 139, 140, 141, 143, 144

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

Equally: numbers n for which A048673(n) is odd.

Also, after 1, numbers n for which A243062(n) is even.

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

Complement: A246263.
Positions of zeros in A246271.
Inverse function: A246262.
The first row of array A246259.
Cf. A003961, A048673, A243062, A246265.

{1}~Join~Select[Range[144], Mod[Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]], 4] == 1 &] (* Michael De Vlieger, Mar 12 2021 *)

from sympy import factorint, prime, primepi
from operator import mul
def a003961(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**f[i] for i in f])
print([n for n in range(1, 201) if a003961(n)%4==1]) # Indranil Ghosh, Jun 12 2017

For all n >= 1, A246262(a(n)) = n.

A246263 Numbers n such that A003961(n) is of the form 4k+3.

Original entry on oeis.org

2, 5, 6, 7, 8, 15, 17, 18, 19, 20, 21, 22, 24, 26, 28, 29, 32, 41, 43, 45, 46, 50, 51, 53, 54, 55, 57, 60, 61, 62, 63, 65, 66, 67, 68, 70, 72, 73, 74, 76, 77, 78, 79, 80, 84, 87, 88, 91, 94, 96, 98, 101, 103, 104, 112, 113, 115, 116, 118, 123, 125, 127, 128, 129, 135, 137, 138, 142, 149, 150

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

Equally: numbers n for which A048673(n) is even.
Also, after the first term (2), numbers n for which A243062(n) is odd.
Numbers that occur below the topmost row of square array A246259.

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

Complement: A246261.
Positions of nonzeros in A246271.
Inverse function: A246264.
Cf. A003961, A048673, A246259, A243062, A246267.

A243061 Permutation of natural numbers, a composition of A241909 and A064216: a(n) = A064216(A241909(n)).

Original entry on oeis.org

1, 2, 5, 3, 6, 13, 29, 4, 7, 47, 20, 25, 113, 95, 15, 11, 78, 23, 355, 158, 103, 267, 406, 89, 19, 1247, 17, 1237, 1577, 139, 660, 10, 221, 4363, 67, 38, 8179, 13109, 967, 393, 9266, 515, 21605, 4162, 28, 23601, 19578, 239, 43, 83, 987, 31247

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

This is A241909-conjugate of A243065. Please see the comments at the latter sequence.

Inverse permutation: A243062.

Cf. A064216, A241909, A243065-A243066.

A064216(n) = A064989(n+n-1);
A064989(n) = { my(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) };
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A243061(n) = A064216(A241909(n)); \\ Antti Karttunen, Dec 10 2021

(define (A243061 n) (A064216 (A241909 n)))

a(n) = A064216(A241909(n)).

a(n) = A241909(A243065(A241909(n))).

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].

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A243065 Permutation of natural numbers, the odd bisection of A241909 halved; equally, a composition of A064216 and A241909: a(n) = A241909(A064216(n)).

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A243066 Permutation of natural numbers, the even bisection of A241909 incremented by one and halved; equally, a composition of A241909 and A048673: a(n) = A048673(A241909(n)).

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A246261 Numbers n such that A003961(n) is of the form 4k+1.

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A246263 Numbers n such that A003961(n) is of the form 4k+3.

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A243061 Permutation of natural numbers, a composition of A241909 and A064216: a(n) = A064216(A241909(n)).

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