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

1,2

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

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.

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)

Example

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.

Mathematica

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

Prog

(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

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).

Sequence in context: A035149 A391239 A074848 * A365173 A390555 A366991

Adjacent sequences: A252502 A252503 A252504 * A252506 A252507 A252508

Keyword

nonn,easy,mult

Author

Geoffrey Critzer, Mar 21 2015

Status

approved