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

1,4

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).

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.

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)

Example

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.

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 *)

Prog

(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

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.

Sequence in context: A085731 A131301 A350698 * A366245 A083730 A358272

Adjacent sequences: A335321 A335322 A335323 * A335325 A335326 A335327

Keyword

nonn,easy,mult

Author

Peter Munn, May 31 2020

Status

approved