A252735 a(1) = 0; for n > 1: a(2n) = a(n), a(2n+1) = 1 + a(A064989(n)).

0, 0, 1, 0, 2, 1, 3, 0, 1, 2, 4, 1, 5, 3, 2, 0, 6, 1, 7, 2, 3, 4, 8, 1, 2, 5, 1, 3, 9, 2, 10, 0, 4, 6, 3, 1, 11, 7, 5, 2, 12, 3, 13, 4, 2, 8, 14, 1, 3, 2, 6, 5, 15, 1, 4, 3, 7, 9, 16, 2, 17, 10, 3, 0, 5, 4, 18, 6, 8, 3, 19, 1, 20, 11, 2, 7, 4, 5, 21, 2, 1, 12, 22, 3, 6, 13, 9, 4, 23, 2, 5, 8, 10, 14, 7, 1, 24, 3, 4, 2, 25, 6, 26, 5, 3, 15, 27, 1

Offset: 1

Author

Antti Karttunen, Dec 21 2014

Keywords

• nonn

Comments

Consider the binary tree illustrated in A005940: If we start from any n, computing successive iterations of A252463 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

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Essentially one less than A061395.

Cf. A252464, A252736.

Cf. A000120, A005940, A064989, A080791, A156552, A243071, A252462, A252464, A252736.

Cf. also A246369.

Programs

Mathematica

a252735[n_] := Prepend[Rest@Array[PrimePi[FactorInteger[#][[-1]][[1]]] - 1 &, n], 0]; a252735[108] (* Michael De Vlieger, Dec 21 2014, after Stefan Steinerberger at A061395 *)

Formula

a(1) = 0; for n > 1: a(2n) = a(n), a(2n+1) = 1 + a(A064989(n)).

a(n) = A080791(A156552(n)). [Number of nonleading 0-bits in A156552(n).]

Other identities:

For all n >= 2:

a(n) = A061395(n) - 1.

a(n) = A000120(A243071(n)) - 1. [One less than the binary weight of A243071(n).]

a(n) = A252464(n) - A252736(n) - 1.