cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Antti Karttunen, Jan 12 2015

Keywords

Comments

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

Links

Crossrefs

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.

Programs

Formula

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)

Extensions

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

A252754 Inverse of "Tree of Eratosthenes" permutation: a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A268674(2n+1)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 128, 19, 14, 33, 256, 23, 12, 65, 18, 35, 512, 21, 1024, 31, 22, 129, 20, 27, 2048, 257, 34, 39, 4096, 29, 8192, 67, 30, 513, 16384, 47, 24, 25, 26, 131, 32768, 37, 28, 71, 38, 1025, 65536, 43, 131072, 2049, 66, 63, 36, 45, 262144, 259, 46, 41, 524288, 55, 1048576, 4097, 130, 515, 40
Offset: 1

Views

Author

Antti Karttunen, Jan 02 2015

Keywords

Links

Crossrefs

Inverse: A252753.
Fixed points of a(n)+1: A253789.
Cf. A250470, A253556, A253557, A253558, A253559, A268674, A302041.
Similar permutations: A156552, A252756, A054429, A250246, A269388.
Differs from A156552 for the first time at n=21, where a(21) = 14, while A156552(21) = 18.

Programs

Formula

a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A268674(2n+1)).
As a composition of related permutations:
a(n) = A054429(A252756(n)).
a(n) = A156552(A250246(n)).
From Antti Karttunen, Mar 31 2018: (Start)
A000120(a(n)) = A253557(n).
A069010(a(n)) = A302041(n).
A132971(a(n)) = A302050(n).
A106737(a(n)) = A302051(n).
(End)

Extensions

Name edited and formula corrected by Antti Karttunen, Mar 31 2018

A253555 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A250470(2n+1)); also binary width of terms of A252754 and A252756.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 12 2015

Keywords

Comments

a(n) tells how many iterations of A253554 are needed before 1 is reached, i.e., the distance of n from 1 in binary trees like A252753 and A252755.

Links

Crossrefs

Cf. A000040, A000079, A001248, A253554.
Cf. also A252753, A252754, A252755, A252756, A253557, A253558, A253559.
Differs from A252464 for the first time at n=21, where a(21) = 4, while A252463(21) = 5.

Formula

a(1) = 0; for n > 1: a(n) = 1 + a(A253554(n)).
a(n) = A029837(1+A252754(n)) = A029837(1+A252756(n)).
a(n) = A253556(n) + A253557(n).
Other identities.
For all n >= 1:
a(A000079(n)) = n. [I.e., a(2^n) = n.]
a(A000040(n)) = n.
a(A001248(n)) = n+1.
For n >= 2, a(n) = A253558(n) + A253559(n).

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

Views

Author

Antti Karttunen, Jan 12 2015

Keywords

Comments

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).

Links

Crossrefs

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.

Formula

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

Views

Author

Antti Karttunen, Jan 12 2015

Keywords

Comments

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).

Links

Crossrefs

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.

Programs

Formula

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).
Showing 1-5 of 5 results.

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