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

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

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.

Sequence in context: A286605 A035149 A074848 * A365173 A366991 A365680

Adjacent sequences: A252502 A252503 A252504 * A252506 A252507 A252508

Keyword

nonn,easy,mult

Author

Geoffrey Critzer, Mar 21 2015

Status

approved