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

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

Author

Antti Karttunen, Jan 12 2015

Keywords

• nonn

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

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

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

Scheme

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

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