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

1,4

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

Links

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

Formula

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

Example

   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

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

Prog

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

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.

Sequence in context: A128428 A056171 A357328 * A238949 A076755 A317751

Adjacent sequences: A333746 A333747 A333748 * A333750 A333751 A333752

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 03 2020

Status

approved