A252505 Number of biquadratefree (4th power free) divisors of n.

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

Offset: 1

Author

Geoffrey Critzer, Mar 21 2015

Keywords

• nonn
• easy
• mult

Comments

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

Examples

			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.
		

References

• Paul J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 37, Exercise 1.27.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Jon Maiga, Computer-generated formulas for A252505, Sequence Machine.
• Eric Weisstein's World of Mathematics, Biquadratefree.

Crossrefs

Cf. A046100 (biquadratefree numbers).

Cf. A034444 (squarefree divisors), A073184 (cubefree divisors).

Cf. A001620.

Cf. A000005, A058035, A307430.

Also obtained as a Dirichlet convolution of the following pairs: A034444 and A227291, A007427 and A286779, A008966 and A323308, A048691 and A363552, A271102 and A322327, A307445 and A370296, and A018892 and A378214 (conjectured).

Programs

Mathematica

Prepend[Table[Apply[Times, (FactorInteger[n][[All, 2]] /. x_ /; x > 3 -> 3) + 1], {n, 2, 100}], 1]

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

Formula

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

a(n) = Sum_{d|n} mu(gcd(d, n/d))^2. - Ilya Gutkovskiy, Feb 21 2020

Multiplicative with a(p^e) = min(e, 3) + 1. - Amiram Eldar, Sep 19 2020

From Antti Karttunen, May 14 2025: (Start)

Following formulas have been generated for this sequence by Sequence Machine:

a(n) = A000005(A058035(n)).

a(n) = Sum_{d|n} A307430(d).

a(n) = Sum_{d|n} A034444(d)*A227291(n/d).

a(n) = Sum_{d|n} A007427(d)*A286779(n/d).

a(n) = Sum_{d|n} A008966(d)*A323308(n/d).

a(n) = Sum_{d|n} A048691(d)*A363552(n/d).

a(n) = Sum_{d|n} A271102(d)*A322327(n/d).

a(n) = Sum_{d|n} A307445(d)*A370296(n/d).

a(n) = Sum_{d|n} A018892(d)*A378214(n/d). [Conjectured]

(End)