A250249 -id:A250249 - 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

A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

If n >= 2, n occurs in column a(n) of A246278.

By convention, a(1) = 0 because 1 does not occur in A246278.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Terms of A348717 halved. A305897 is the restricted growth sequence transform.
Positions of terms 1 .. 8 in this sequence are given by the following sequences: A000040, A001248, A006094, A030078, A090076, A251720, A090090, A030514.
Cf. A078898 (has the same role with array A083221 as this sequence has with A246278).
Cf. A000040, A001222, A001358, A003961, A055396, A064989, A064216, A243055, A246272, A249810, A249820, A249735, A252463.
This sequence is also used in the definition of the following permutations: A246274, A246276, A246675, A246677, A246683, A249815, A249817 (A249818), A249823, A249825, A250244, A250245, A250247, A250249.
Also in the definition of arrays A249821, A251721, A251722.
Sum of prime indices of a(n) is A359358(n) + A001222(n) - 1, cf. A326844.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A005940, A241916, A253565, A356958, A358195.

a246277[n_Integer] := Module[{f, p, a064989, 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]]];
  a[1] = 0;
  a[x_] := If[EvenQ[x], x/2, NestWhile[a064989, x, OddQ]/2];
a/@Range[n]]; a246277[84] (* Michael De Vlieger, Dec 19 2014 *)

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)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2); \\ Antti Karttunen, Apr 30 2022

from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a064989(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])
def a(n): return 0 if n==1 else n//2 if n%2==0 else a(a064989(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

;; two different variants, the second one employing memoizing definec-macro)
(define (A246277 n) (if (= 1 n) 0 (let loop ((n n)) (if (even? n) (/ n 2) (loop (A064989 n))))))
(definec (A246277 n) (cond ((= 1 n) 0) ((even? n) (/ n 2)) (else (A246277 (A064989 n)))))

a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)) = a(A064216(n+1)). [Cf. the formula for A252463.]
Instead of the equation for a(2n+1) above, we may write a(A003961(n)) = a(n). - Peter Munn, May 21 2022
Other identities. For all n >= 1, the following holds:
For all w >= 0, a(p_{i} * p_{j} * ... * p_{k}) = a(p_{i+w} * p_{j+w} * ... * p_{k+w}).
For all n >= 2, A001222(a(n)) = A001222(n)-1. [a(n) has one less prime factor than n. Thus each semiprime (A001358) is mapped to some prime (A000040), etc.]
For all n >= 2, a(n) = A078898(A249817(n)).
For semiprimes n = p_i * p_j, j >= i, a(n) = A000040(1+A243055(n)) = p_{1+j-i}.
a(n) = floor(A348717(n)/2). - Antti Karttunen, Apr 30 2022
If n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=2..k} prime(x_i-x_1+1). The opposite version is A358195, prime indices A358172, even bisection A241916. - Gus Wiseman, Dec 29 2022

A083221 Sieve of Eratosthenes arranged as an array and read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 21, 35, 49, 11, 12, 27, 55, 77, 121, 13, 14, 33, 65, 91, 143, 169, 17, 16, 39, 85, 119, 187, 221, 289, 19, 18, 45, 95, 133, 209, 247, 323, 361, 23, 20, 51, 115, 161, 253, 299, 391, 437, 529, 29, 22, 57, 125, 203, 319, 377, 493, 551, 667

Offset: 2

Yasutoshi Kohmoto, Jun 05 2003

This is permutation of natural numbers larger than 1.
From Antti Karttunen, Dec 19 2014: (Start)
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252460 gives an inverse permutation. See also A249741.
For navigating in this array:
A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.
A250469(n) gives the number immediately below n, and when n is an odd number >= 3, A250470(n) gives the number immediately above n. If n is a composite, A249744(n) gives the number immediately left of n.
First cube of each row, which is {the initial prime of the row}^3 and also the first number neither a prime or semiprime, occurs on row n at position A250474(n).
(End)
The n-th row contains the numbers whose least prime factor is the n-th prime: A020639(T(n,k)) = A000040(n). - Franklin T. Adams-Watters, Aug 07 2015

			The top left corner of the array:
   2,   4,   6,    8,   10,   12,   14,   16,   18,   20,   22,   24,   26
   3,   9,  15,   21,   27,   33,   39,   45,   51,   57,   63,   69,   75
   5,  25,  35,   55,   65,   85,   95,  115,  125,  145,  155,  175,  185
   7,  49,  77,   91,  119,  133,  161,  203,  217,  259,  287,  301,  329
  11, 121, 143,  187,  209,  253,  319,  341,  407,  451,  473,  517,  583
  13, 169, 221,  247,  299,  377,  403,  481,  533,  559,  611,  689,  767
  17, 289, 323,  391,  493,  527,  629,  697,  731,  799,  901, 1003, 1037
  19, 361, 437,  551,  589,  703,  779,  817,  893, 1007, 1121, 1159, 1273
  23, 529, 667,  713,  851,  943,  989, 1081, 1219, 1357, 1403, 1541, 1633
  29, 841, 899, 1073, 1189, 1247, 1363, 1537, 1711, 1769, 1943, 2059, 2117
  ...

Transpose of A083140.
One more than A249741.
Inverse permutation: A252460.
Column 1: A000040, Column 2: A001248.
Row 1: A005843, Row 2: A016945, Row 3: A084967, Row 4: A084968, Row 5: A084969, Row 6: A084970.
Main diagonal: A083141.
First semiprime in each column occurs at A251717; A251718 & A251719 with additional criteria. A251724 gives the corresponding semiprimes for the latter. See also A251728.
Permutations based on mapping numbers between this array and A246278: A249817, A249818, A250244, A250245, A250247, A250249. See also: A249811, A249814, A249815.
Also used in the definition of the following arrays of permutations: A249821, A251721, A251722.
Cf. A002260, A004736, A004280, A020639, A038179, A055396, A078898, A138511, A249820, A249730, A249735, A249744, A250469, A250470, A250472, A250474.

lim = 11; a = Table[Take[Prime[n] Select[Range[lim^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], lim], {n, lim}]; Flatten[Table[a[[i, n - i + 1]], {n, lim}, {i, n}]] (* Michael De Vlieger, Jan 04 2016, after Yasutoshi Kohmoto at A083140 *)

More terms from Hugo Pfoertner, Jun 13 2003

A078442 a(p) = a(n) + 1 if p is the n-th prime, prime(n); a(n)=0 if n is not prime.

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Henry Bottomley, Dec 31 2002

nonn

Fernandez calls this the order of primeness of n.
a(A007097(n))=n, for any n >= 0. - Paul Tek, Nov 12 2013
When a nonoriented rooted tree is encoded as a Matula-Goebel number n, a(n) tells how many edges needs to be climbed up from the root of the tree until the first branching vertex (or the top of the tree, if n is one of the terms of A007097) is encountered. Please see illustrations at A061773. - Antti Karttunen, Jan 27 2014
Zero-based column index of n in the Kimberling-style dispersion table of the primes (see A114537). - Allan C. Wechsler, Jan 09 2024

			a(1) = 0 since 1 is not prime;
a(2) = a(prime(1)) = a(1) + 1 = 1 + 0 = 1;
a(3) = a(prime(2)) = a(2) + 1 = 1 + 1 = 2;
a(4) = 0 since 4 is not prime;
a(5) = a(prime(3)) = a(3) + 1 = 2 + 1 = 3;
a(6) = 0 since 6 is not prime;
a(7) = a(prime(4)) = a(4) + 1 = 0 + 1 = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
Index entries for sequences related to Matula-Goebel numbers

A left inverse of A007097.
One less than A049076.
a(A000040(n)) = A049076(n).
Cf. A373338 (mod 2), A018252 (positions of zeros).
Cf. A000720, A049084, A061773.
Cf. permutations A235489, A250247/A250248, A250249/A250250, A245821/A245822 that all preserve a(n).
Cf. also array A114537 (A138947) and permutations A135141/A227413, A246681.

a078442 n = fst $ until ((== 0) . snd)
                        (\(i, p) -> (i + 1, a049084 p)) (-2, a000040 n)
-- Reinhard Zumkeller, Jul 14 2013

A078442 := proc(n)
    if not isprime(n) then
        0 ;
    else
        1+procname(numtheory[pi](n)) ;
    end if;
end proc: # R. J. Mathar, Jul 07 2012

a[n_] := a[n] = If[!PrimeQ[n], 0, 1+a[PrimePi[n]]]; Array[a, 105] (* Jean-François Alcover, Jan 26 2018 *)

A078442(n)=for(i=0,n, isprime(n) || return(i); n=primepi(n)) \\ M. F. Hasler, Mar 09 2010

a(n) = A049076(n)-1.
a(n) = if A049084(n) = 0 then 0 else a(A049084(n)) + 1. - Reinhard Zumkeller, Jul 14 2013
For all n, a(n) = A007814(A135141(n)) and a(A227413(n)) = A007814(n). Also a(A235489(n)) = a(n). - Antti Karttunen, Jan 27 2014

A249817 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),A246277(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 42, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 54, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 66, 67, 68, 111, 70, 71, 72, 73, 74, 51, 76, 77, 78, 79, 80, 45, 82, 83, 84, 155, 86, 135

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

a(n) tells which number in square array A083221 (the sieve of Eratosthenes) is at the same position where n is in array A246278. As both arrays have even numbers as their topmost row and primes as their leftmost column, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A083140 is at the same position where n is in the array A246279, as they are the transposes of above two arrays.

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

Inverse: A249818.
There are three different "deep" versions of this permutation, recursing on values of A055396(n) and/or A246277(n), namely: A250245, A250247 and A250249.
Other similar or related permutations: A249815.
Cf. A000040, A005843, A020639, A055396, A078898, A083140, A083221, A246277, A246278, A246279, A249821.
Differs from its inverse A249818 for the first time at n=33, where a(33) = 39, while A249818(33) = 45.

lim = 87; a083221 = Table[Take[Prime[n] Select[Range[Ceiling[lim/2]^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], Ceiling[lim/2]], {n, Ceiling[lim/2]}]; a055396[n_] PrimePi[FactorInteger[n][[1, 1]]]; a246277[n_] := Which[n == 1, 0, EvenQ@ n, n/2, True, a246277[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ Transpose@ FactorInteger@ n, Last@ Transpose@ FactorInteger@ n]]]; Table[a083221[[a055396@ n, a246277@ n]], {n, 2, lim}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A055396 and Yasutoshi Kohmoto at A083140 *)

(define (A249817 n) (if (= 1 n) n (A083221bi (A055396 n) (A246277 n)))) ;; Code for A083221bi given in A083221
;; Alternative version:
(define (A249817 n) (if (= 1 n) n (A083221bi (A055396 n) (A249821bi (A055396 n) (A078898 n))))) ;; Code for A249821bi given in A249821.

a(1) = 1, a(n) = A083221(A055396(n), A246277(n)).
a(1) = 1, a(n) = A083221(A055396(n), A249821(A055396(n), A078898(n))).
As a composition of other permutations:
a(1) = 1, and for n > 1, a(n) = 1 + A249815(n-1).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n) and a(A000040(n)) = A000040(n). [Fixes even numbers and primes, among other numbers. Cf. comments above].
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].

A250245 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),a(A246277(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 54, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 42, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 78, 67, 68, 111, 70, 71, 72, 73, 74, 51, 76, 77, 126, 79, 80, 45, 82

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

The first 7-cycle occurs at: (33 39 63 57 99 81 45) which is mirrored by the cycle (66 78 126 114 198 162 90) with double-size terms.
The cycle which contains 55 as its smallest term, goes as: 55, 65, 95, 185, 425, 325, 205, 455, 395, 1055, 2945, 6035, 30845, ...
while to the other direction (A250246) it goes as: 55, 125, 245, 115, 625, 8575, 40375, ...
The cycle which contains 69 as its smallest term, goes as: 69, 111, 183, 351, 261, 273, 387, 489, 939, 1863, 909, 1161, 981, 1281, 4167, ...
while to the other direction (A250246) it goes as: 69, 135, 87, 105, 225, 207, 231, 195, 525, 1053, 3159, 24909, ...

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

Inverse: A250246.
Other similar permutations: A250244, A250247, A250249, A243071, A252755.
Cf. A005843, A020639, A055396, A064989, A083221, A246277, A250469.
Differs from the "vanilla version" A249817 for the first time at n=42, where a(42) = 54, while A249817(42) = 42.
Differs from A250246 for the first time at n = 33, where a(33) = 39, while A250246(33) = 45.
Differs from A250249 for the first time at n=73, where a(73) = 73, while A250249(73) = 103.

a(1) = 1, a(n) = A083221(A055396(n), a(A246277(n))).
a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A064989(2n+1))). - Antti Karttunen, Jan 18 2015
As a composition of related permutations:
a(n) = A252755(A243071(n)).
Other identities. For all n >= 1:
a(n) = a(2n)/2. [The even bisection halved gives the sequence back.]
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].

A250250 Permutation of natural numbers: a(1) = 1, a(n) = A246278(a(A055396(n)),a(A078898(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 103, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

This is a "doubly-recursed" version of A249818.

Antti Karttunen, Table of n, a(n) for n = 1..3432
Index entries for sequences that are permutations of the natural numbers

Inverse: A250249.
Fixed points: A250251, their complement: A249729.
Cf. A000035, A000040, A055396, A078442, A078898, A049076, A246278.
See also other (somewhat) similar permutations: A245821, A057505.
Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.

a(1) = 1, a(n) = A246278(a(A055396(n)), a(A078898(n))).
Other identities. For all n >= 1:
a(2n) = 2*a(n), or equally, a(n) = a(2n)/2. [The even bisection halved gives the sequence back].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].
A000035(n) = A000035(a(n)). [Preserves the parity].

A250247 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)),A246277(n)).

Original entry on oeis.org

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

The first 7-cycle occurs at: (33 39 63 57 99 81 45), which is mirrored at the cycle (137 167 307 269 523 419 197), consisting of primes (p_33, p_39, p_63, ...).

			As a(21) = 27, and A000040(21) = 73 and A000040(27) = 103, a(73) = 103.

Antti Karttunen, Table of n, a(n) for n = 1..4240
Index entries for sequences that are permutations of the natural numbers

Inverse: A250248.
Cf. A000040, A005843, A049076, A049084, A055396, A078442, A083221, A246277.
Differs from its inverse A250248 for the first time at n = 33, where a(33) = 39, while A250248(33) = 45.
Differs from the "vanilla version" A249817 for the first time at n=73, where a(73) = 103, while A249817(73) = 73.
Differs from "doubly recursed" version A250249 for the first time at n=42, where a(42) = 42, while A250249(42) = 54, thus the first prime where they get different values is p_42 = 181, where a(181) = 181, while A250249(181) = 251 = p_54.

a(1) = 1, a(n) = A083221(a(A055396(n)),A246277(n)).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n). [Fixes even numbers].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].

Numbers for which A250249(n) <> n (equally: A250250(n) <> n).

If n is a member, then 2n is also a member. If any 2n is a member, then n is also a member. If n is a member, then the n-th prime, p_n (= A000040(n)) is also a member. If p_n is a member, then its index n is also a member. Thus the sequence is completely determined by its odd nonprime terms: 21, 27, 33, 39, 45, ..., and is obtained as a union of their multiples with powers of 2, and all prime recurrences that start with those values. For example, because 21 is present, then 2*21 = 42 is also present. Furthermore, 73 = p_21 is also present, as well as 367 = p_73 as well as 181 = p_42. See also comments at A250251 and A250249.

Numbers for which A250249(n) = n (equally: A250250(n) = n).

If n is a member, then 2n is also a member. If any 2n is a member, then n is also a member. If n is a member, then the n-th prime, p_n (= A000040(n)) is also a member. If p_n is a member, then its index n is also a member. Thus the sequence is completely determined by its odd nonprime terms: 1, 9, 15, 25, ..., (A249730) and is obtained as a union of their multiples with powers of 2, and all prime recurrences that start with those values: A007097 U A057450 U A057451 U A057452 U A057453 U ..., etc.

cp's OEIS Frontend

A249729 Numbers not fixed by A250249 and A250250.

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A250251 Fixed points of A250249 and A250250.

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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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A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

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A083221 Sieve of Eratosthenes arranged as an array and read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A078442 a(p) = a(n) + 1 if p is the n-th prime, prime(n); a(n)=0 if n is not prime.

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A249817 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),A246277(n)).

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