A358434 Number of odd middle divisors of n, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

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

Offset

1,15

Comments

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

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

Links

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

Formula

a(n) = A067742(n) - A361561(n). - Omar E. Pol, Mar 31 2023

Example

For n = 8 the middle divisor of 8 is [2]. There are no odd middle divisors of 8 so a(8) = 0.
For n = 12 the middle divisors of 12 are [3, 4]. There is only one odd middle divisor of 12 so a(12) = 1.
For n = 15 the middle divisors of 15 are [3, 5]. There are two odd middle divisors of 15 so a(15) = 2.

Mathematica

Table[DivisorSum[n, 1 &, And[OddQ[#], Sqrt[n/2] <= # < Sqrt[2*n]] &], {n, 120}] (* Michael De Vlieger, Mar 31 2023 *)

Prog

(PARI) a(n) = #select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2)) && x%2), divisors(n)); \\ Michel Marcus, Mar 26 2023

Crossrefs

Cf. A001227, A067742, A071090, A071562, A299761, A361561.

Sequence in context: A116376 A165766 A102082 * A030199 A320005 A325414

Adjacent sequences: A358431 A358432 A358433 * A358435 A358436 A358437

Keyword

nonn

Author

Omar E. Pol, Mar 14 2023

Status

approved