A161906 Triangle read by rows in which row n lists the divisors of n that are <= sqrt(n).

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 3, 1, 2, 4, 1, 1, 2, 3, 1, 1, 2, 4, 1, 3, 1, 2, 1, 1, 2, 3, 4, 1, 5, 1, 2, 1, 3, 1, 2, 4, 1, 1, 2, 3, 5, 1, 1, 2, 4, 1, 3, 1, 2, 1, 5, 1, 2, 3, 4, 6, 1, 1, 2, 1, 3, 1, 2, 4, 5, 1, 1, 2, 3, 6, 1, 1, 2, 4, 1, 3, 5, 1, 2, 1, 1, 2, 3

Offset: 1

Author

Omar E. Pol, Jun 27 2009

Keywords

• easy
• nonn
• tabf

Comments

If we define a divisor d|n to be inferior if d <= n/d, then inferior divisors are counted by A038548 and listed by this sequence. - Gus Wiseman, Mar 08 2021

Examples

			Triangle begins:
   1....... 1;
   2....... 1;
   3....... 1;
   4..... 1,2;
   5....... 1;
   6..... 1,2;
   7....... 1;
   8..... 1,2;
   9..... 1,3;
  10..... 1,2;
  11....... 1;
  12... 1,2,3;
  13....... 1;
  14..... 1,2;
  15..... 1,3;
  16... 1,2,4;
		

Links

• Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Crossrefs

Initial terms are A000012.

Final terms are A033676.

Row lengths are A038548 (number of inferior divisors).

Row sums are A066839 (sum of inferior divisors).

The prime terms are counted by A063962.

The odd terms are counted by A069288.

Row products are A072499.

Row LCMs are A072504.

The superior version is A161908.

The squarefree terms are counted by A333749.

The prime-power terms are counted by A333750.

The strictly superior version is A341673.

The strictly inferior version is A341674.

A001221 counts prime divisors, with sum A001414.

A000005 counts divisors, listed by A027750 with sum A000203.

A056924 count strictly superior (or strictly inferior divisors).

A207375 lists central divisors.

- Inferior: A217581.

- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A341591, A341592, A341593, A341675, A341676.

- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341677.

- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646.

Cf. A000196, A001055, A001248, A006530, A020639, A050320, A068101, A161901.

Programs

Haskell

a161906 n k = a161906_tabf !! (n-1) !! (k-1)
a161906_row n = a161906_tabf !! (n-1)
a161906_tabf = zipWith (\m ds -> takeWhile ((<= m) . (^ 2)) ds)
                       [1..] a027750_tabf'
-- Reinhard Zumkeller, Jun 24 2015, Mar 08 2013

Mathematica

div[n_] := Select[Divisors[n], # <= Sqrt[n] &]; div /@ Range[48] // Flatten (* Amiram Eldar, Nov 13 2020 *)

PARI

row(n) = select(x->(x<=sqrt(n)), divisors(n)); \\ Michel Marcus, Nov 13 2020

Extensions

More terms from Sean A. Irvine, Nov 29 2010