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A252505 Number of biquadratefree (4th power free) divisors of n. 2
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 (list; graph; refs; listen; history; text; internal format)
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.

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

CROSSREFS

Cf. A046100 (biquadratefree numbers).

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

Sequence in context: A286605 A035149 A074848 * A325560 A318412 A322986

Adjacent sequences:  A252502 A252503 A252504 * A252506 A252507 A252508

KEYWORD

nonn,mult

AUTHOR

Geoffrey Critzer, Mar 21 2015

STATUS

approved

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Last modified December 8 10:20 EST 2021. Contains 349594 sequences. (Running on oeis4.)