A252755 -id:A252755 - cp's OEIS frontend

A163511 a(0)=1. a(n) = p(A000120(n)) * Product_{m=1..A000120(n)} p(m)^A163510(n,m), where p(m) is the m-th prime.

1, 2, 4, 3, 8, 9, 6, 5, 16, 27, 18, 25, 12, 15, 10, 7, 32, 81, 54, 125, 36, 75, 50, 49, 24, 45, 30, 35, 20, 21, 14, 11, 64, 243, 162, 625, 108, 375, 250, 343, 72, 225, 150, 245, 100, 147, 98, 121, 48, 135, 90, 175, 60, 105, 70, 77, 40, 63, 42, 55, 28, 33, 22, 13, 128

Offset: 0

Leroy Quet, Jul 29 2009

This is a permutation of the positive integers.
From Antti Karttunen, Jun 20 2014: (Start)
Note the indexing: the domain starts from 0, while the range excludes zero, thus this is neither a bijection on the set of nonnegative integers nor on the set of positive natural numbers, but a bijection from the former set to the latter.
Apart from that discrepancy, this could be viewed as yet another "entanglement permutation" where the two complementary pairs to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with the complementary pair even numbers (taken straight) and odd numbers in the order they appear in A003961: (A005843/A003961). See also A246375 which has almost the same recurrence.
Note how the even bisection halved gives the same sequence back. (For a(0)=1, take ceiling of 1/2).
(End)
From Antti Karttunen, Dec 30 2017: (Start)
This irregular table can be represented as a binary tree. Each child to the left is obtained by doubling the parent, and each child to the right is obtained by applying A003961 to the parent:
                                     1
                                     |
                  ...................2...................
                 4                                       3
       8......../ \........9                   6......../ \........5
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  16       27         18       25         12       15         10       7
32  81   54  125    36  75   50  49     24  45   30  35     20  21   14 11
etc.
Sequence A005940 is obtained by scanning the same tree level by level in mirror image fashion. Also in binary trees A253563 and A253565 the terms on level of the tree are some permutation of the terms present on the level n of this tree. A252464(n) gives the distance of n from 1 in all these trees, and A252463 gives the parent of the node containing n.
A252737(n) gives the sum and A252738(n) the product of terms on row n (where 1 is on row 0, 1 on row 1, 3 and 4 on row 2, etc.). A252745(n) gives the number of nodes on level n whose left child is smaller than the right child, and A252744(n) is an indicator function for those nodes.
(End)
Note that the idea behind maps like this (and the mirror image A005940) admits also using alternative orderings of primes, not just standard magnitude-wise ordering (A000040). For example, A332214 is a similar sequence but with primes rearranged as in A332211, and A332817 is obtained when primes are rearranged as in A108546. - Antti Karttunen, Mar 11 2020
From Lorenzo Sauras Altuzarra, Nov 28 2020: (Start)
This sequence is generated from A228351 by applying the following procedure: 1) eliminate the compositions that end in one unless the first one, 2) subtract one unit from every component, 3) replace every tuple [t_1, ..., t_r] by Product_{k=1..r} A000040(k)^(t_k) (see the examples).
Is it true that a(n) = A337909(n+1) if and only if a(n+1) is not a term of A161992?
Does this permutation have any other cycle apart from (1), (2) and (6, 9, 16, 7)? (End)
From Antti Karttunen, Jul 25 2023: (Start)
(In the above question, it is assumed that the starting offset would be 1 instead of 0).
Questions:
Does a(n) = 1+A054429(n) hold only when n is of the form 2^k times 1, 3 or 7, i.e., one of the terms of A029748?
It seems that A007283 gives all fixed points of map n -> a(n), like A335431 seems to give all fixed points of map n -> A332214(n). Is there a general rule for mappings like these that the fixed points (if they exist) must be of the form 2^k times a certain kind of prime, i.e., that any odd composite (times 2^k) can certainly be excluded? See also note in A029747.
(End)
If the conjecture given in A364297 holds, then it implies the above conjecture about A007283. See also A364963. - Antti Karttunen, Sep 06 2023
Conjecture: a(n^k) is never of the form x^k, for any integers n > 0, k > 1, x >= 1. This holds at least for squares, cubes, seventh and eleventh powers (see A365808, A365801, A366287 and A366391). - Antti Karttunen, Sep 24 2023, Oct 10 2023.
See A365805 for why the above holds for any n^k, with k > 1. - Antti Karttunen, Nov 23 2023

			For n=3, whose binary representation is "11", we have A000120(3)=2, with A163510(3,1) = A163510(3,2) = 0, thus a(3) = p(2) * p(1)^0 * p(2)^0 = 3*1*1 = 3.
For n=9, "1001" in binary, we have A000120(9)=2, with A163510(9,1) = 0 and A163510(9,2) = 2, thus a(9) = p(2) * p(1)^0 * p(2)^2 = 3*1*9 = 27.
For n=10, "1010" in binary, we have A000120(10)=2, with A163510(10,1) = 1 and A163510(10,2) = 1, thus a(10) = p(2) * p(1)^1 * p(2)^1 = 3*2*3 = 18.
For n=15, "1111" in binary, we have A000120(15)=4, with A163510(15,1) = A163510(15,2) = A163510(15,3) = A163510(15,4) = 0, thus a(15) = p(4) * p(1)^0 * p(2)^0 * p(3)^0 * p(4)^0 = 7*1*1*1*1 = 7.
[1], [2], [1,1], [3], [1,2], [2,1] ... -> [1], [2], [3], [1,2], ... -> [0], [1], [2], [0,1], ... -> 2^0, 2^1, 2^2, 2^0*3^1, ... = 1, 2, 4, 3, ... - _Lorenzo Sauras Altuzarra_, Nov 28 2020

Inverse: A243071.
Cf. A000040, A000120, A000225, A000788, A003961, A007814, A054429, A055396, A064216, A135523, A161992, A163510, A245605, A245612, A246375, A246378, A246681, A161511, A228351, A243499, A243503, A243504, A269854, A280873, A285727, A290251, A293437, A337909.
Cf. A007283 (known positions where a(n)=n), A029747, A029748, A364255 [= gcd(n,a(n))], A364258 [= a(n)-n], A364287 (where a(n) < n), A364292 (where a(n) <= n), A364494 (where n|a(n)), A364496 (where a(n)|n), A364963, A364297.
Cf. A365808 (positions of squares), A365801 (of cubes), A365802 (of fifth powers), A365805 [= A052409(a(n))], A366287, A366391.
Cf. A005940, A332214, A332817, A366275 (variants).

f[n_] := Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]]; {1}~Join~
Table[Function[t, Prime[t] Product[Prime[m]^(f[n][[m]]), {m, t}]][DigitCount[n, 2, 1]], {n, 120}] (* Michael De Vlieger, Jul 25 2016 *)

from sympy import prime
def A163511(n):
    if n:
        k, c, m = n, 0, 1
        while k:
            c += 1
            m *= prime(c)**(s:=(~k&k-1).bit_length())
            k >>= s+1
        return m*prime(c)
    return 1 # Chai Wah Wu, Jul 17 2023

For n >= 1, a(2n) is even, a(2n+1) is odd. a(2^k) = 2^(k+1), for all k >= 0.
From Antti Karttunen, Jun 20 2014: (Start)
a(0) = 1, a(1) = 2, a(2n) = 2*a(n), a(2n+1) = A003961(a(n)).
As a more general observation about the parity, we have:
For n >= 1, A007814(a(n)) = A135523(n) = A007814(n) + A209229(n). [This permutation preserves the 2-adic valuation of n, except when n is a power of two, in which cases that value is incremented by one.]
For n >= 1, A055396(a(n)) = A091090(n) = A007814(n+1) + 1 - A036987(n).
For n >= 1, a(A000225(n)) = A000040(n).
(End)
From Antti Karttunen, Oct 11 2014: (Start)
As a composition of related permutations:
a(n) = A005940(1+A054429(n)).
a(n) = A064216(A245612(n))
a(n) = A246681(A246378(n)).
Also, for all n >= 0, it holds that:
A161511(n) = A243503(a(n)).
A243499(n) = A243504(a(n)).
(End)
More linking identities from Antti Karttunen, Dec 30 2017: (Start)
A046523(a(n)) = A278531(n). [See also A286531.]
A278224(a(n)) = A285713(n). [Another filter-sequence.]
A048675(a(n)) = A135529(n) seems to hold for n >= 1.
A250245(a(n)) = A252755(n).
A252742(a(n)) = A252744(n).
A245611(a(n)) = A253891(n).
A249824(a(n)) = A275716(n).
A292263(a(n)) = A292264(n). [A292944(n) + A292264(n) = n.]
--
A292383(a(n)) = A292274(n).
A292385(a(n)) = A292271(n). [A292271(n) +  A292274(n) = n.]
--
A292941(a(n)) = A292942(n).
A292943(a(n)) = A292944(n).
A292945(a(n)) = A292946(n). [A292942(n) + A292944(n) + A292946(n) = n.]
--
A292253(a(n)) = A292254(n).
A292255(a(n)) = A292256(n). [A292944(n) + A292254(n) + A292256(n) = n.]
--
A279339(a(n)) = A279342(n).
a(A071574(n)) = A269847(n).
a(A279341(n)) = A279338(n).
a(A252756(n)) = A250246(n).
(1+A008836(a(n)))/2 = A059448(n).
(End)
From Antti Karttunen, Jul 26 2023: (Start)
For all n >= 0, a(A007283(n)) = A007283(n).
A001222(a(n)) = A290251(n).
(End)

More terms computed and examples added by Antti Karttunen, Jun 20 2014

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].

A253557 a(1) = 0; after which, a(2n) = 1 + a(n), a(2n+1) = a(A268674(2n+1)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 2, 2, 2, 3, 1, 3, 1, 5, 3, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 3, 4, 2, 1, 5, 2, 3, 3, 3, 1, 3, 3, 4, 3, 2, 1, 4, 1, 2, 2, 6, 2, 4, 1, 3, 4, 3, 1, 5, 1, 2, 2, 3, 2, 3, 1, 5, 3, 2, 1, 5, 3, 2, 3, 4, 1, 5, 3, 3, 5, 2, 2, 6, 1, 3, 2, 4, 1, 4, 1, 4, 4, 2, 1, 4, 1, 4, 2, 5, 1

Offset: 1

Antti Karttunen, Jan 12 2015

nonn

Consider the binary trees illustrated in A252753 and A252755: If we start from any n, computing successive iterations of A253554 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located), a(n) gives the number of even numbers encountered on the path (i.e., including both 2 and the starting n if it was even).

This is bigomega (A001222) analog for nonstandard factorization based on the sieve of Eratosthenes (A083221). See A302041 for an omega-analog. - Antti Karttunen, Mar 31 2018

Antti Karttunen, Table of n, a(n) for n = 1..8192

Essentially, one more than A253559.
Primes, A000040, gives the positions of ones.
Cf. A000079, A000120, A080791, A252753, A252754, A252755, A252756, A253554, A253555, A253556, A253558, A302037, A302041, A302042.
Differs from A001222 for the first time at n=21, where a(21) = 3, while A001222(21) = 2.

(definec (A253557 n) (cond ((= 1 n) 0) ((odd? n) (A253557 (A250470 n))) (else (+ 1 (A253557 (/ n 2))))))

a(1) = 0; after which, a(2n) = 1 + a(n), a(2n+1) = a(A268674(2n+1)).
a(n) = A253555(n) - A253556(n).
a(n) = A000120(A252754(n)). [Binary weight of A252754(n).]
Other identities.
For all n >= 0:
a(2^n) = n.
For all n >= 2:
a(n) = A080791(A252756(n)) + 1. [One more than the number of nonleading 0-bits in A252756(n).]
From Antti Karttunen, Apr 01 2018: (Start)
a(1) = 0; for n > 1, a(n) = 1 + a(A302042(n)).
a(n) = A001222(A250246(n)).
(End)

Definition (formula) corrected by Antti Karttunen, Mar 31 2018

A252753 Tree of Eratosthenes: a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 21, 16, 11, 14, 27, 20, 35, 30, 33, 24, 49, 50, 51, 36, 55, 42, 45, 32, 13, 22, 39, 28, 65, 54, 57, 40, 77, 70, 87, 60, 85, 66, 69, 48, 121, 98, 147, 100, 125, 102, 105, 72, 91, 110, 123, 84, 115, 90, 93, 64, 17, 26, 63, 44, 95, 78, 81, 56, 119, 130, 159, 108, 145, 114, 117, 80

Offset: 0

Antti Karttunen, Jan 02 2015

This sequence can be represented as a binary tree. Each child to the left is obtained by applying A250469 to the parent, and each child to the right is obtained by doubling the parent:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........6                   9......../ \........8
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  7       10         15       12         25       18         21       16
11 14   27  20     35  30   33  24     49  50   51  36     55  42   45  32
etc.
Sequence A252755 is the mirror image of the same tree. A253555(n) gives the distance of n from 1 in both trees.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A252754.
Row sums: A253787, products: A253788.
Fixed points of a(n-1): A253789.
Similar permutations: A005940, A252755, A054429, A250245.
Cf. also A001248, A001511, A055396, A083221, A181565, A250469, A253555.

(* b = A250469 *)
b[1] = 1; b[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[ 1, 1]]; For[k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1+2 == k2, Return[m2]]]];
a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ[n], b[a[n/2]], 2 a[(n-1)/2]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2016 *)

a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).
As a composition of related permutations:
a(n) = A252755(A054429(n)).
a(n) = A250245(A005940(1+n)).
Other identities. For all n >= 1:
A055396(a(n)) = A001511(n). [A005940 has the same property.]
a(A003945(n)) = A001248(n) for n>=1. - Peter Luschny, Jan 13 2015

A252756 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A250470(2n+1)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 31, 12, 63, 30, 13, 8, 127, 10, 255, 28, 9, 62, 511, 24, 11, 126, 29, 60, 1023, 26, 2047, 16, 25, 254, 27, 20, 4095, 510, 61, 56, 8191, 18, 16383, 124, 17, 1022, 32767, 48, 23, 22, 21, 252, 65535, 58, 19, 120, 57, 2046, 131071, 52, 262143, 4094, 125, 32, 59, 50, 524287, 508, 49, 54, 1048575, 40

Offset: 1

Antti Karttunen, Jan 02 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A252755.
Similar permutations: A243071, A252754, A054429, A250246.
Cf. also A250470, A253556 - A253559.
Differs from A243071 for the first time at n=21, where a(21) = 9, while A243071(21) = 29.

a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A250470(2n+1)).
As a composition of related permutations:
a(n) = A054429(A252754(n)).
a(n) = A243071(A250246(n)).

A249814 "Mountains of Eratosthenes" permutation: a(1) = 1, a(n) = A249741(A001511(n), a(A003602(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 15, 24, 13, 20, 11, 10, 17, 26, 27, 34, 29, 44, 47, 48, 25, 38, 39, 54, 21, 32, 19, 12, 33, 50, 51, 64, 53, 80, 67, 76, 57, 86, 87, 114, 93, 140, 95, 120, 49, 74, 75, 94, 77, 116, 107, 90, 41, 62, 63, 84, 37, 56, 23, 16, 65, 98, 99, 124, 101, 152, 127, 118, 105, 158, 159, 204, 133, 200, 151, 142

Offset: 1

Antti Karttunen, Nov 06 2014

This sequence is a "recursed variant" of A249811.
From Antti Karttunen, Jan 18 2015: (Start)
This can be viewed as an entanglement or encoding permutation where the complementary pairs of sequences to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with another complementary pair: even numbers in the order they appear in A253886 and odd numbers in their usual order: (A253886/A005408).
From the above follows also that this sequence can be represented as a binary tree. Each child to the left is obtained by doubling the parent and subtracting one, and each child to the right is obtained by applying A253886 to the parent:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........8                   7......../ \........6
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  9       14         15       24         13       20         11       10
17 26   27  34     29  44   47  48     25  38   39  54     21  32   19  12
(End)
For listening I recommend some (mostly) percussive MIDI-instrument and the pitch offset set to at least 29 and the tempo (rate) to about 60. - Antti Karttunen, Feb 17 2015

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

Inverse: A249813.
Similar or related permutations: A246684, A249811, A250244, A252755.
Cf. A000079, A000265, A001511, A003602, A006093, A083140, A083221, A249741, A250469, A253886.
Compare also the scatterplot of this sequence to the graphs of A252755 and A246684.
Differs from A246684 for the first time at n=14, where a(14) = 20, while A246684(14) = 26.

In the following formulas, A083221 and A249741 are interpreted as bivariate functions:
a(1) = 1, for n>1: a(n) = A083221(A001511(n), a(A003602(n))) - 1 = A249741(A001511(n), a(A003602(n))).
a(1) = 1, a(2n) = A253886(a(n)), a(2n+1) = (2*a(n+1))-1. - Antti Karttunen, Jan 18 2015
As a composition of other permutations:
a(n) = A250244(A246684(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A000079(n-1)) = A006093(n).

cp's OEIS Frontend

A253787 Row sums of irregular tables A252753 & A252755.

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A253788 Row products of irregular tables A252753 & A252755.

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A286550 a(n) = A252755(n) - n.

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A286551 Restricted growth sequence of A286550 (= A252755(n) - n).

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A163511 a(0)=1. a(n) = p(A000120(n)) * Product_{m=1..A000120(n)} p(m)^A163510(n,m), where p(m) is the m-th prime.

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

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A253557 a(1) = 0; after which, a(2n) = 1 + a(n), a(2n+1) = a(A268674(2n+1)).

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A252753 Tree of Eratosthenes: a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).

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A252756 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A250470(2n+1)).

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A252756 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A250470(2n+1)).