A333749 Number of squarefree divisors of n that are <= sqrt(n).

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

Offset: 1

Author

Ilya Gutkovskiy, Apr 03 2020

Keywords

• nonn

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 A161906. This sequence counts inferior squarefree divisors. - Gus Wiseman, Feb 27 2021

Examples

			   n   inferior squarefree divisors of n
  ---  ---------------------------------
   33  1,  3
   56  1,  2,  7
  429  1,  3, 11, 13
   90  1,  2,  3,  5,  6
  490  1,  2,  5,  7, 10, 14
  480  1,  2,  3,  5,  6, 10, 15
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A034444, A069291, A333748, A333752.

Positions of 1's are A008578.

The case of equality is the indicator function of A062503.

The version for prime instead of squarefree divisors is A063962.

The version for odd instead of squarefree divisors is A069288.

The version for prime-power instead of squarefree divisors is A333750.

The superior version is A341592.

The strictly superior version is A341595.

The strictly inferior version is A341596.

A005117 lists squarefree numbers.

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A161906 lists inferior divisors.

A161908 lists superior divisors.

A207375 list central divisors.

- Inferior: A033676, A066839, A217581.

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

- Strictly Inferior: A060775, A070039, A333805, A333806, A341674, A341677.

- Strictly Superior: A048098, A064052 A140271, A238535, A341591, A341594, A341642, A341643, A341644, A341645/A341646, A341673.

Cf. A000005, A000203, A001221, A001222, A001248, A006530, A020639.

Programs

Maple

N:= 200: # for a(1)..a(N)
g:= add(x^(k^2)/(1-x^k), k = select(numtheory:-issqrfree,[$1..floor(sqrt(N))])):
S:= series(g,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Apr 05 2020

Mathematica

Table[DivisorSum[n, 1 &, # <= Sqrt[n] && SquareFreeQ[#] &], {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = sumdiv(n, d, (d^2<=n) && issquarefree(d)); \\ Michel Marcus, Apr 03 2020

Formula

G.f.: Sum_{k>=1} mu(k)^2 * x^(k^2) / (1 - x^k).