A383032 Exponent of the highest power of 2 dividing the n-th number that is either a square or twice a square.

0, 1, 2, 3, 0, 4, 1, 0, 5, 2, 0, 1, 6, 3, 0, 1, 2, 0, 7, 4, 1, 0, 2, 3, 0, 1, 8, 5, 0, 2, 1, 0, 3, 4, 0, 1, 2, 9, 0, 6, 1, 0, 3, 2, 1, 0, 4, 5, 0, 1, 2, 0, 3, 10, 1, 0, 7, 2, 0, 1, 4, 3, 0, 2, 1, 0, 5, 6, 0, 1, 2, 3, 0, 1, 4, 0, 11, 2, 1, 0, 8, 3, 0, 1, 2, 5, 0

Offset: 1

Author

Amiram Eldar, Apr 13 2025

Keywords

• nonn
• easy

Comments

The first position of k in this sequence, for k >= 0, is A001521(k+1).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001521, A001954, A003151, A003152, A007814, A014176, A028982, A215247, A342280.

Programs

Mathematica

With[{m = 3000}, IntegerExponent[Union[Join[Range[Floor[Sqrt[m]]]^2, 2*Range[Floor[Sqrt[m/2]]]^2]], 2]]

PARI

lista(nn) = apply(x->valuation(x,2), vecsort(concat(vector(sqrtint(nn\1), i, i^2), vector(sqrtint(nn\2), i, 2*i^2)))); \\ Michel Marcus, Apr 13 2025

Formula

a(n) = A007814(A028982(n)).

a(A001954(n)) = 0 for n >= 0.

a(A215247(n)) = 1 for n >= 1.

a(A342280(n)) = 2 for n >= 0.

Sum_{k=1..n} a(k) ~ (1 + sqrt(2)) * n.