A335324 Square part of 4th-power-free part of n.

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 1, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1

Offset: 1

Author

Peter Munn, May 31 2020

Keywords

• nonn
• easy
• mult

Comments

Equivalently, biquadratefree (4th-power-free) part of square part of n.

Multiplicative. The terms are squares of squarefree numbers (A062503).

Every positive integer n is the product of a unique subset S_n of the terms of A050376 (sometimes called Fermi-Dirac primes). a(n) is the product of the members of S_n that are squares of prime numbers (A001248).

Examples

			Removing the 4th powers from 192 = 2^6 * 3^1 gives 2^(6 - 4) * 3^1 = 2^2 * 3 = 12. So the 4th-power-free part of 192 is 12. The square part of 12 (largest square dividing 12) is 4. So a(192) = 4.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Biquadratefree.
• Eric Weisstein's World of Mathematics, Square part.

Crossrefs

A007913, A008833, A008835, A053165 are used in formulas defining the sequence.

Column 1 of A352780.

Range of values is A062503.

Positions of 1's: A252895.

Related to A038500 by A225546.

The formula section details how the sequence maps the terms of A003961, A331590.

Cf. A001248, A050376, A083730.

Cf. A002117, A013662, A078434.

Programs

Mathematica

f[p_, e_] := p^(2*Floor[e/2] - 4*Floor[e/4]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 01 2020 *)

PARI

A053165(n)=my(f=factor(n)); f[, 2]=f[, 2]%4; factorback(f);
a(n) = my(m=A053165(n)); m/core(m); \\ Michel Marcus, Jun 01 2020

Python

from math import prod
from sympy import factorint
def A335324(n): return prod(p**(e&2) for p, e in factorint(n).items()) # Chai Wah Wu, Aug 07 2024

Formula

a(n) = A053165(A008833(n)) = A008833(A053165(n)).

a(n) = A053165(n) / A007913(n).

a(n) = A008833(n) / A008835(n).

n = A007913(n) * a(n) * A008835(n).

a(n) = A225546(A038500(A225546(n))).

a(n^2) = A007913(n)^2.

a(A003961(n)) = A003961(a(n)).

a(A331590(n, k)) = A331590(a(n), a(k)).

a(p^e) = p^(2*floor(e/2) - 4*floor(e/4)). - Amiram Eldar, Jun 01 2020

From Amiram Eldar, Sep 21 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-2) * zeta(4*s)/(zeta(2*s) * zeta(4*s-4)).

Sum_{k=1..n} a(k) ~ (4*zeta(3/2)*zeta(4))/(21*zeta(3)) * n^(3/2). (End)