A341596 Number of strictly inferior squarefree divisors of n.

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 3, 2, 1, 4, 1, 3, 2, 2, 1, 4, 2, 3, 2, 2, 1, 5, 1, 2, 3, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 3, 2, 2, 4, 1, 3, 2, 2, 1, 5, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Feb 23 2021

Keywords

• nonn

Comments

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.

Examples

			The strictly inferior squarefree divisors of selected n:
n = 1  2  6  12  30  60  120  210  240  420  630  1050  1260
    --------------------------------------------------------
    {} 1  1  1   1   1   1    1    1    1    1    1     1
          2  2   2   2   2    2    2    2    2    2     2
             3   3   3   3    3    3    3    3    3     3
                 5   5   5    5    5    5    5    5     5
                     6   6    6    6    6    6    6     6
                         10   7    10   7    7    7     7
                              10   15   10   10   10    10
                              14        14   14   14    14
                                        15   15   15    15
                                             21   21    21
                                                  30    30
                                                        35
		

Links

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

Crossrefs

Positions of ones are A000430.

The weakly inferior version is A333749.

The version counting odd instead of squarefree divisors is A333805.

The version counting prime instead of squarefree divisors is A333806.

The weakly superior version is A341592.

The strictly superior version is A341595.

The version counting prime-power instead of squarefree divisors is A341677.

A001221 counts prime divisors, with sum A001414.

A001222 counts prime power divisors.

A005117 lists squarefree numbers.

A033676 selects the greatest inferior divisor.

A033677 selects the smallest superior divisor.

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A207375 lists central divisors.

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

- Superior: A051283, A059172, A063538, A063539, A070038, A116882, A116883, A161908, A341591, A341593, A341675, A341676.

- Strictly Inferior: A060775, A070039, A341674.

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

Cf. A000005, A000203, A001055, A001248, A050320.

Programs

Mathematica

Table[Length[Select[Divisors[n],SquareFreeQ[#]&&#
				

PARI

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