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Number of squarefree divisors of n that are <= sqrt(n).
30

%I #20 Mar 02 2021 09:21:31

%S 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,

%T 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,

%U 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

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

%C 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

%H Robert Israel, <a href="/A333749/b333749.txt">Table of n, a(n) for n = 1..10000</a>

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

%e n inferior squarefree divisors of n

%e --- ---------------------------------

%e 33 1, 3

%e 56 1, 2, 7

%e 429 1, 3, 11, 13

%e 90 1, 2, 3, 5, 6

%e 490 1, 2, 5, 7, 10, 14

%e 480 1, 2, 3, 5, 6, 10, 15

%p N:= 200: # for a(1)..a(N)

%p g:= add(x^(k^2)/(1-x^k), k = select(numtheory:-issqrfree,[$1..floor(sqrt(N))])):

%p S:= series(g,x,N+1):

%p seq(coeff(S,x,j),j=1..N); # _Robert Israel_, Apr 05 2020

%t Table[DivisorSum[n, 1 &, # <= Sqrt[n] && SquareFreeQ[#] &], {n, 1, 100}]

%t nmax = 100; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%o (PARI) a(n) = sumdiv(n, d, (d^2<=n) && issquarefree(d)); \\ _Michel Marcus_, Apr 03 2020

%Y Cf. A034444, A069291, A333748, A333752.

%Y Positions of 1's are A008578.

%Y The case of equality is the indicator function of A062503.

%Y The version for prime instead of squarefree divisors is A063962.

%Y The version for odd instead of squarefree divisors is A069288.

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

%Y The superior version is A341592.

%Y The strictly superior version is A341595.

%Y The strictly inferior version is A341596.

%Y A005117 lists squarefree numbers.

%Y A038548 counts superior (or inferior) divisors.

%Y A056924 counts strictly superior (or strictly inferior) divisors.

%Y A161906 lists inferior divisors.

%Y A161908 lists superior divisors.

%Y A207375 list central divisors.

%Y - Inferior: A033676, A066839, A217581.

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

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

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

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

%K nonn

%O 1,4

%A _Ilya Gutkovskiy_, Apr 03 2020