A341592 Number of squarefree superior divisors of n.

1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 0, 2, 1, 4, 1, 0, 2, 2, 2, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 0, 1, 1, 2, 2, 1, 0, 2, 1, 2, 2, 1, 3, 1, 2, 1, 0, 2, 4, 1, 2, 2, 4, 1, 0, 1, 2, 1, 2, 2, 4, 1, 1, 0, 2, 1, 3, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Feb 19 2021

Keywords

• nonn

Comments

We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.

Examples

			The strictly superior squarefree divisors (columns) of selected n:
  1  6  8  30  60  210  420  630  1050  2310  4620  6930
  ------------------------------------------------------
  1  3  .   6  10   15   21   30   35     55    70   105
     6     10  15   21   30   35   42     66    77   110
           15  30   30   35   42   70     70   105   154
           30       35   42   70  105     77   110   165
                    42   70  105  210    105   154   210
                    70  105  210         110   165   231
                   105  210              154   210   330
                   210                   165   231   385
                                         210   330   462
                                         231   385   770
                                         330   462  1155
                                         385   770  2310
                                         462  1155
                                         770  2310
                                        1155
                                        2310
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of zeros are A059172.

The inferior version is A333749.

The version for prime instead of squarefree divisors is A341591.

The version for prime powers instead of squarefree divisors is A341593.

The strictly superior case is A341595.

The version for odd instead of squarefree divisors is A341675.

A001221 counts prime divisors, with sum A001414.

A033677 selects the smallest superior divisor.

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A161908 lists superior divisors.

A207375 lists central divisors.

- Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333750.

- Superior: A051283, A063538, A063539, A070038, A116882, A116883, A341676.

- Strictly Inferior: A060775, A333805, A333806, A341596, A341674.

- Strictly Superior: A048098, A064052 A140271, A238535, A341594, A341642, A341643, A341644, A341645, A341646, A341673.

Cf. A000005, A000203, A001222, A001248, A006530, A020639, A112798.

Programs

Maple

with(numtheory):
a := n -> nops(select(d -> d*d >= n and issqrfree(d), divisors(n))):
seq(a(n), n = 1..88); # Peter Luschny, Feb 20 2021

Mathematica

Table[Length[Select[Divisors[n],SquareFreeQ[#]&&#>=n/#&]],{n,100}]

PARI

a(n) = sumdiv(n, d, d^2 >= n && issquarefree(d)); \\ Amiram Eldar, Nov 01 2024