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

Author

Omar E. Pol, Mar 15 2023

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

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

Programs

Mathematica

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

PARI

a(n) = sumdiv(n, d, if (!(d%2), (d>=sqrt(n/2)) && (dMichel Marcus, Mar 15 2023

Formula

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