

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.


22



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
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OFFSET

1,4


COMMENTS

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


LINKS



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

The number of nonzero terms in row n is A067742(n).
Indices of the rows where there are zeros give A071561.
Indices of the rows where there are nonzero terms give A071562.


KEYWORD



AUTHOR



STATUS

approved



