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A333749
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Number of squarefree divisors of n that are <= sqrt(n).
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30
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} mu(k)^2 * x^(k^2) / (1 - x^k).
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EXAMPLE
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n inferior squarefree divisors of n
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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
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MAPLE
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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):
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MATHEMATICA
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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
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PROG
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(PARI) a(n) = sumdiv(n, d, (d^2<=n) && issquarefree(d)); \\ Michel Marcus, Apr 03 2020
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CROSSREFS
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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 strictly superior version is A341595.
The strictly inferior version is A341596.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A341593, A341675, A341676.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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