A341645 Numbers without a strictly superior squarefree divisor.

1, 4, 8, 9, 16, 25, 27, 32, 36, 48, 49, 54, 64, 72, 81, 96, 100, 108, 121, 125, 128, 144, 160, 162, 169, 192, 196, 200, 216, 224, 225, 243, 250, 256, 288, 289, 320, 324, 343, 361, 375, 384, 392, 400, 405, 432, 441, 448, 484, 486, 500, 512, 529, 567, 576, 625

Offset: 1

Author

Gus Wiseman, Feb 22 2021

Keywords

• nonn

Comments

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.

Examples

			72 has strictly superior divisors {9,12,18,24,36,72} and squarefree divisors {1,2,3,6}, but the intersection is empty, so 72 is in the sequence.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

The version for prime instead of squarefree divisors is A048098.

The version for prime-power instead of squarefree divisors is A051283.

The weakly superior version is A059172.

The version for odd instead of squarefree divisors is A116882.

Positions of zeros in A341595.

The complement is A341646.

A001221 counts prime divisors, with sum A001414.

A005117 lists squarefree numbers.

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A140271 selects the smallest strictly superior divisor.

A207375 list central divisors.

A341673 lists strictly superior divisors.

- Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.

- Superior: A033677, A063538, A063539, A070038, A072500, A116883, A161908, A341591, A341592, A341593, A341675, A341676.

- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341674.

- Strictly Superior: A064052, A238535, A341594, A341642, A341643, A341644.

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

Programs

Mathematica

Select[Range[100],Function[n,Select[Divisors[n],SquareFreeQ[#]&&#>n/#&]=={}]]

PARI

isok(m) = my(d=divisors(m)); #select(x->(issquarefree(x) && (x^2>m)), d) == 0; \\ Michel Marcus, Feb 11 2024