A335324 - OEIS

%I #24 Aug 07 2024 15:11:58

%S 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,

%T 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,

%U 4,1,1,1,36,1,1,25,4,1,1,1,1,1,1,1,4,1

%N Square part of 4th-power-free part of n.

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

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

%C 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).

%H Michel Marcus, <a href="/A335324/b335324.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Biquadratefree.html">Biquadratefree</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarePart.html">Square part</a>.

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

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

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

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

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

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

%F a(A003961(n)) = A003961(a(n)).

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

%F a(p^e) = p^(2*floor(e/2) - 4*floor(e/4)). - _Amiram Eldar_, Jun 01 2020

%F From _Amiram Eldar_, Sep 21 2023: (Start)

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

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

%e 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.

%t 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 *)

%o (PARI) A053165(n)=my(f=factor(n)); f[, 2]=f[, 2]%4; factorback(f);

%o a(n) = my(m=A053165(n)); m/core(m); \\ _Michel Marcus_, Jun 01 2020

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A335324(n): return prod(p**(e&2) for p, e in factorint(n).items()) # _Chai Wah Wu_, Aug 07 2024

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

%Y Column 1 of A352780.

%Y Range of values is A062503.

%Y Positions of 1's: A252895.

%Y Related to A038500 by A225546.

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

%Y Cf. A001248, A050376, A083730.

%Y Cf. A002117, A013662, A078434.

%K nonn,easy,mult

%O 1,4

%A _Peter Munn_, May 31 2020