A361824 Sum 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, 3, 0, 0, 3, 0, 0, 3, 0, 0, 8, 0, 0, 3, 0, 5, 0, 0, 0, 0, 5, 0, 0, 7, 0, 5, 0, 0, 0, 0, 12, 0, 0, 0, 0, 5, 0, 7, 0, 0, 14, 0, 0, 0, 7, 5, 0, 0, 0, 9, 0, 7, 0, 0, 0, 0, 0, 0, 16, 0, 0, 11, 0, 0, 0, 7, 0, 9, 0, 0, 0, 0, 18, 0, 0, 0, 9, 0, 0, 7, 0, 0, 0, 11, 0, 9, 20, 0, 0, 0, 0, 0, 0, 7, 20, 0

Offset

1,6

Comments

Sum of odd divisors of n in the half-open interval [sqrt(n/2), sqrt(n*2)) (cf. A067742).

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

Links

Winston de Greef, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Michael De Vlieger, Detail of scalar scatterplot of a(n), n = 1..2^16, a(n) <= 384.

Formula

a(n) = A071090(n) - A361879(n).

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) = 3.
For n = 15 the middle divisors of 15 are [3, 5]. There are two odd middle divisors of 15 so a(15) = 3 + 5 = 8.

Mathematica

Table[Total@ Select[Divisors[n], And[Sqrt[n/2] <= # < Sqrt[2 n], OddQ[#] ] &], {n, 100}] (* Michael De Vlieger, Mar 27 2023 *)

Prog

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

Crossrefs

Cf. A000203, A000593, A067742, A071090, A071562, A299761, A299777, A303297, A358434, A361879.

Sequence in context: A346607 A095104 A021337 * A293903 A284444 A341794

Adjacent sequences: A361821 A361822 A361823 * A361825 A361826 A361827

Keyword

nonn,look

Author

Omar E. Pol, Mar 25 2023

Status

approved