A361561 Number of even middle divisors of n, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset

1,24

Comments

Number of even divisors of n in the half-open interval [sqrt(n/2), sqrt(n*2)).

Also number of even numbers in the n-th row of A299761.

Links

Table of n, a(n) for n=1..105.

Formula

a(n) = A067742(n) - A358434(n).

Example

For n = 18 the middle divisor of 18 is [3]. There are no even middle divisors of 18 so a(18) = 0.
For n = 20 the middle divisors of 20 are [4, 5]. There is only one even middle divisor of 20 so a(20) = 1.
For n = 24 the middle divisors of 24 are [4, 6]. There are two even middle divisors of 24 so a(24) = 2.

Mathematica

a[n_] := Count[Divisors[n], _?(EvenQ[#] && Sqrt[n/2] <= # < Sqrt[2*n] &)]; Array[a, 100] (* Amiram Eldar, Mar 16 2023 *)

Prog

(PARI) a(n) = sumdiv(n, d, if (!(d%2), (d>=sqrt(n/2)) && (d<sqrt(2*n)))); \\ Michel Marcus, Mar 15 2023

Crossrefs

Cf. A067742, A071090, A071562, A183063, A299761, A358434, A361824.

Sequence in context: A088183 A308470 A070140 * A081212 A280751 A280749

Adjacent sequences: A361558 A361559 A361560 * A361562 A361563 A361564

Keyword

nonn

Author

Omar E. Pol, Mar 15 2023

Status

approved