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A252505
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Number of biquadratefree (4th power free) divisors of n.
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5
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1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 8, 2, 6, 6, 9
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OFFSET
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1,2
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COMMENTS
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Equivalently, a(n) is the number of divisors of n that are in A046100.
a(n) is also the number of divisors d such that the greatest common square divisor of d and n/d is 1.
The number of divisors d of n such that gcd(d, n/d) is squarefree. - Amiram Eldar, Aug 25 2023
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REFERENCES
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Paul J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 37, Exercise 1.27.
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LINKS
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FORMULA
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Dirichlet g.f.: zeta(s)^2/zeta(4*s).
Sum_{k=1..n} a(k) ~ 90*n/Pi^4 * (log(n) - 1 + 2*gamma - 360*zeta'(4)/Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019
Multiplicative with a(p^e) = min(e, 3) + 1. - Amiram Eldar, Sep 19 2020
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EXAMPLE
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a(16) = 4 because there are 4 divisors of 16 that are 4th power free: 1,2,4,8.
a(16) = 4 because there are 4 divisors d of 16 such that the greatest common square divisor of d and 16/d is 1: 1,2,8,16.
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MATHEMATICA
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Prepend[Table[Apply[Times, (FactorInteger[n][[All, 2]] /. x_ /; x > 3 -> 3) + 1], {n, 2, 100}], 1]
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PROG
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(PARI) isA046100(n) = (n==1) || vecmax(factor(n)[, 2])<4;
a(n) = {d = divisors(n); sum(i=1, #d, isA046100(d[i])); } \\ Michel Marcus, Mar 22 2015
(PARI) a(n) = vecprod(apply(x->min(x, 3) + 1, factor(n)[, 2])); \\ Amiram Eldar, Aug 25 2023
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CROSSREFS
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Cf. A046100 (biquadratefree numbers).
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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