A299761 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which row n lists the middle divisors of n, or 0 if there are no middle divisors of n.

1, 1, 0, 2, 0, 2, 3, 0, 2, 3, 0, 0, 3, 4, 0, 0, 3, 5, 4, 0, 3, 0, 4, 5, 0, 0, 0, 4, 6, 5, 0, 0, 4, 7, 0, 5, 6, 0, 4, 0, 0, 5, 7, 6, 0, 0, 0, 5, 8, 0, 6, 7, 0, 0, 5, 9, 0, 0, 6, 8, 7, 5, 0, 0, 0, 6, 9, 0, 7, 8, 0, 0, 0, 6, 10, 0, 0, 7, 9, 8, 0, 6, 11, 0, 0, 0, 7, 10, 0, 6, 8, 9, 0, 0, 0, 0, 7, 11, 0, 0, 8, 10

Offset

1,4

Comments

The middle divisors of n are the divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..14002 (rows 1 <= n <= 10^4)

Example

Triangle begins (rows 1..16):
1;
1;
0;
2;
0;
2, 3;
0;
2;
3;
0;
0;
3, 4;
0;
0;
3, 5;
4;
...
For n = 6 the middle divisors of 6 are 2 and 3, so row 6 is [2, 3].
For n = 7 there are no middle divisors of 7, so row 7 is [0].
For n = 8 the middle divisor of 8 is 2, so row 8 is [2].
For n = 72 the middle divisors of 72 are 6, 8 and 9, so row 72 is [6, 8, 9].

Mathematica

Table[Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] /. {} -> {0}, {n, 80}] // Flatten (* Michael De Vlieger, Jun 14 2018 *)

Prog

(PARI) row(n) = my(v=select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2))), divisors(n))); if (#v, v, [0]); \\ Michel Marcus, Aug 04 2022

Crossrefs

Row sums give A071090.

The number of nonzero terms in row n is A067742(n).

Nonzero terms give A303297.

Indices of the rows where there are zeros give A071561.

Indices of the rows where there are nonzero terms give A071562.

Cf. A027750, A281007, A299777.

Sequence in context: A209689 A204329 A111565 * A141099 A127710 A137510

Adjacent sequences: A299758 A299759 A299760 * A299762 A299763 A299764

Keyword

nonn,tabf,look

Author

Omar E. Pol, Jun 08 2018

Status

approved