A253555 -id:A253555 - cp's OEIS frontend

A252755 Tree of Eratosthenes, mirrored: a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A250469(a(n)).

1, 2, 4, 3, 8, 9, 6, 5, 16, 21, 18, 25, 12, 15, 10, 7, 32, 45, 42, 55, 36, 51, 50, 49, 24, 33, 30, 35, 20, 27, 14, 11, 64, 93, 90, 115, 84, 123, 110, 91, 72, 105, 102, 125, 100, 147, 98, 121, 48, 69, 66, 85, 60, 87, 70, 77, 40, 57, 54, 65, 28, 39, 22, 13, 128, 189, 186, 235, 180, 267, 230, 203, 168, 249, 246, 305, 220, 327, 182, 187, 144

Offset: 0

Antti Karttunen, Jan 02 2015

This sequence 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 A250469 to the parent:
                                     1
                                     |
                  ...................2...................
                 4                                       3
       8......../ \........9                   6......../ \........5
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  16       21         18       25         12       15         10       7
32  45   42  55     36  51   50  49     24  33   30  35     20  27   14 11
etc.
Sequence A252753 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
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences that are permutations of the natural numbers

Inverse: A252756.
Row sums: A253787, products: A253788.
Similar permutations: A163511, A252753, A054429, A163511, A250245, A269865.
Cf. also: A249814 (Compare the scatterplots).
Cf. A083221, 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], 2 a[n/2], b[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) = 2*a(n), a(2n+1) = A250469(a(n)).
As a composition of related permutations:
a(n) = A252753(A054429(n)).
a(n) = A250245(A163511(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

A253554 a(1) = 1, a(2n) = n, a(2n+1) = A250470(2n+1).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 4, 5, 7, 6, 11, 7, 6, 8, 13, 9, 17, 10, 8, 11, 19, 12, 9, 13, 10, 14, 23, 15, 29, 16, 12, 17, 15, 18, 31, 19, 14, 20, 37, 21, 41, 22, 16, 23, 43, 24, 25, 25, 18, 26, 47, 27, 21, 28, 20, 29, 53, 30, 59, 31, 22, 32, 27, 33, 61, 34, 24, 35, 67, 36, 71, 37, 26, 38, 35, 39, 73, 40, 28, 41, 79, 42, 33, 43, 30

Offset: 1

Antti Karttunen, Jan 12 2015

nonn

Divide the even numbers by two, and for odd numbers n >= 3, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1.

For any number n >= 2 in binary trees A252753 and A252755, a(n) gives the number which is the parent of n.

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

Bisections: A000027 and A250472.
Cf. A253555 (the number of iterations needed to reach 1 from n).
Cf. A055396, A078898, A250470, A252753, A252755.
Differs from A252463 for the first time at n=21, where a(21) = 8, while A252463(21) = 10.

(define (A253554 n) (cond ((<= n 1) n) ((even? n) (/ n 2)) (else (A250470 n))))

a(1) = 1, a(2n) = n, a(2n+1) = A250470(2n+1).

A253556 a(1) = 0; after which, a(2n) = a(n), a(2n+1) = 1 + a(A250470(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 12 2015

nonn

Consider the binary tree 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 at), a(n) gives the number of odd numbers > 1 encountered on the path (i.e., excluding the final 1 from the count but including the starting n if it was odd).

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

One less than A253558.
Powers of two, A000079, gives the positions of zeros.
Cf. A000120, A080791, A252753, A252754, A252755, A252756, A253554, A253555, A253557, A253559.
Differs from A252735 for the first time at n=21, where a(21) = 1, while A252735(21) = 3.

a(1) = 0; after which, a(2n) = a(n), a(2n+1) = 1 + a(A250470(n)).
a(n) = A253555(n) - A253557(n).
a(n) = A253558(n) - 1.
a(n) = A080791(A252754(n)). [Number of nonleading 0-bits in A252754(n).]
Other identities. For all n >= 2:
a(n) = A000120(A252756(n)) - 1. [One less than the binary weight of A252756(n).]

A253559 a(1) = 0; for n>1: a(n) = A253557(n) - 1.

Original entry on oeis.org

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

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 > 2 encountered on the path (i.e., excluding the 2 from the count but including the starting n if it was even).

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

Essentially, one less than A253557.
A008578 gives the positions of zeros.
Cf. A000120, A080791, A252753, A252754, A252755, A252756, A253554, A253555, A253556, A253558.
Differs from A252736 for the first time at n=21, where a(21) = 2, while A252736(21) = 1.

(define (A253559 n) (if (= 1 n) 0 (+ -1 (A253557 n))))

a(n) = A080791(A252756(n)). [Number of nonleading 0-bits in A252756(n).]
a(1) = 0; for n>1: a(n) = A253557(n) - 1.
Other identities. For all n >= 2:
a(n) = A000120(A252754(n)) - 1. [One less than the binary weight of A252754(n).]
a(n) = A253555(n) - A253558(n).

A253558 a(n) = A253556(n) + 1.

Original entry on oeis.org

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

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 at), a(n) gives the number of odd numbers encountered on the path (i.e., including both the final 1 and the starting n if it was odd).

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

One more than A253556.
Powers of two, A000079, gives the positions of ones.
Cf. A000040, A253555, A253557, A253559.
After n=1, differs from A061395 for the first time at n=21, where a(21) = 2, while A061395(21) = 4.

Scheme
```
(define (A253558 n) (+ 1 (A253556 n)))
```

a(n) = A253556(n) + 1.
a(n) = A080791(A252754(n)) + 1. [One more than the number of nonleading 0-bits in A252754(n).]
Other identities.
For all n >= 1:
a(A000040(n)) = n.
For all n >= 2:
a(n) = A000120(A252756(n)). [Binary weight of A252756(n).]
a(n) = A253555(n) - A253559(n).

cp's OEIS Frontend

A252755 Tree of Eratosthenes, mirrored: a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A250469(a(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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A253554 a(1) = 1, a(2n) = n, a(2n+1) = A250470(2n+1).

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A253556 a(1) = 0; after which, a(2n) = a(n), a(2n+1) = 1 + a(A250470(n)).

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A253559 a(1) = 0; for n>1: a(n) = A253557(n) - 1.

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A253558 a(n) = A253556(n) + 1.

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