A204455 Squarefree product of all odd primes dividing n, and 1 if n is a power of 2: A099985/2.

1, 1, 3, 1, 5, 3, 7, 1, 3, 5, 11, 3, 13, 7, 15, 1, 17, 3, 19, 5, 21, 11, 23, 3, 5, 13, 3, 7, 29, 15, 31, 1, 33, 17, 35, 3, 37, 19, 39, 5, 41, 21, 43, 11, 15, 23, 47, 3, 7, 5, 51, 13, 53, 3, 55, 7, 57, 29, 59, 15, 61, 31, 21, 1, 65, 33, 67, 17, 69, 35, 71, 3

Offset: 1

Author

Wolfdieter Lang, Jan 19 2012

Keywords

• nonn
• mult

Comments

There are no odd primes dividing n iff n is a power of 2.

This sequence coincides with the bisection of A007947 (even indices), which is A099985, dividing out the even prime 2 in the squarefree kernel.

a(n) divides A106609(n) for n>=1. - Alexander R. Povolotsky, Apr 06 2015

Examples

			a(5)=5 because 5 is a single odd prime.
a(9)=3 because 9=3*3 has as squarefree part 3.
a(1)=1 because 1 is a power of 2, having no odd primes as a factor.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A099985, A099984, A007947, A000265, A065463, A106609.

Programs

Maple

A204455 := proc(n)
    local p;
    numtheory[factorset](n) minus {2} ;
    mul(p,p=%) ;
end proc:
seq(A204455(n),n=1..40) ; # R. J. Mathar, Jan 25 2017

Mathematica

f[n_] := Select[First /@ FactorInteger@ n, PrimeQ@ # && OddQ@ # &]; Times @@@ (f /@ Range@ 120) (* Michael De Vlieger, Apr 08 2015 *)

PARI

a(n) = {my(f = factor(n)); prod(k=1, #f~, if (f[k,1] % 2, f[k,1], 1));} \\ Michel Marcus, Apr 07 2015

PARI

a(n) = factorback(setminus(factorint(n)[, 1]~, [2])) \\ Jianing Song, Aug 09 2022

Formula

a(n) = A099985(n)/2 = A007947(2*n)/2.

a(n) = A000265(A007947(n)) = A007947(A000265(n)). - Charles R Greathouse IV, Jan 19 2012

Multiplicative with a(p^e)=p for p <> 2 and a(2^e)=1. - R. J. Mathar, Jul 02 2013

a(n) = Sum_{d|n} phi(d)*mu(2d)^2. - Ridouane Oudra, Sep 02 2019

From Richard L. Ollerton, May 09 2021: (Start)

a(n) = Sum_{k=1..n} mu(2*n/gcd(n,k))^2.

a(n) = Sum_{k=1..n} mu(2*gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

Sum_{k=1..n} a(k) ~ c * n^2, where c = (2/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (2/5) * A065463 = 0.281776... . - Amiram Eldar, Nov 19 2022

a(n) = Sum_{d divides n, d odd} mu(d)^2 * phi(d). - Peter Bala, Feb 01 2024