A303297 List of middle divisors: for every positive integer that has middle divisors, add its middle divisors to the sequence.

1, 1, 2, 2, 3, 2, 3, 3, 4, 3, 5, 4, 3, 4, 5, 4, 6, 5, 4, 7, 5, 6, 4, 5, 7, 6, 5, 8, 6, 7, 5, 9, 6, 8, 7, 5, 6, 9, 7, 8, 6, 10, 7, 9, 8, 6, 11, 7, 10, 6, 8, 9, 7, 11, 8, 10, 9, 7, 12, 8, 11, 9, 10, 7, 13, 8, 12, 7, 9, 11, 10, 8, 13, 9, 12, 10, 11, 8, 14, 9, 13, 8, 10, 12, 15, 11, 9, 14, 8, 10, 13, 11, 12, 9, 15

Offset

1,3

Comments

The middle divisors of k (see A299761) are the divisors in the half-open interval [sqrt(k/2), sqrt(k*2)), k >= 1.

Links

Michel Marcus, Table of n, a(n) for n = 1..6934

Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 2^16.

Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 345, labeling n, and showing composite prime powers in gold, squarefree composites in green, numbers neither squarefree nor composite in blue, and highlighting products of composite prime powers in large light blue.

Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 2^16, with same color function as above so as to show patterns according to prime power decomposition of n.

Example

The middle divisor of 1 is 1, so a(1) = 1.
The middle divisor of 2 is 1, so a(2) = 1.
There are no middle divisors of 3.
The middle divisor of 4 is 2, so a(3) = 2.
There are no middle divisors of 5.
The middle divisors of 6 are 2 and 3, so a(4) = 2 and a(5) = 3.
There are no middle divisors of 7.
The middle divisor of 8 is 2, so a(6) = 2.
The middle divisor of 9 is 3, so a(7) = 3.
There are no middle divisors of 10.
There are no middle divisors of 11.
The middle divisors of 12 are 3 and 4, so a(8) = 3 and a(9) = 4.

Mathematica

Table[Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] /. {} -> Nothing, {n, 135}] // Flatten (* Michael De Vlieger, Jun 14 2018 *)

Prog

(PARI) lista(nn) = {my(list = List()); for (n=1, nn, my(v = select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2))), divisors(n))); for (i=1, #v, listput(list, v[i])); ); Vec(list); } \\ Michel Marcus, Mar 26 2023

Crossrefs

Concatenate the nonzero rows of A299761 (that is, the nonzero terms of A299761).

Cf. A027750, A067742, A071090, A071562, A281007, A299777.

Sequence in context: A358635 A366615 A092333 * A107452 A349198 A205018

Adjacent sequences: A303294 A303295 A303296 * A303298 A303299 A303300

Keyword

nonn,look

Author

Omar E. Pol, Apr 30 2018

Status

approved