Not yet in OEIS: Product of odd divisors of n?

[Jonathan Post]

Is it worth pointing out that:

a(n) = PRODUCT{h == 1 mod 4, and h | n}* PRODUCT{i == 3 mod 4, and i | n}.
a(n) = PRODUCT{j == 1 mod 6, and j | n}* PRODUCT{k == 5 mod 6, and k | n}.

The first of these might be said to use the Gaussian core of n (the
product of Gaussian primes, A002145, of the form 4i+3 dividing n) and
the Pythagorean core of n (the product of Pythagorean primes, A002144,
of the form 4h+1 dividing n). Are these in OEIS?

-- Jonathan Vos Post




jvp> From seqfan-owner at ext.jussieu.fr  Thu Jun 26 21:41:54 2008
jvp> Date: Thu, 26 Jun 2008 12:40:50 -0700
jvp> From: "Jonathan Post" <jvospost3 at gmail.com>
jvp> To: "David Wilson" <davidwwilson at comcast.net>,
jvp>         "Vladeta Jovovic" <vladeta at eunet.yu>, SeqFan <seqfan at ext.jussieu.fr>
jvp> Subject: Re: Not yet in OEIS: Product of odd divisors of n?
jvp> 
jvp> Is it worth pointing out that:
jvp> 
jvp> a(n) = PRODUCT{h == 1 mod 4, and h | n}* PRODUCT{i == 3 mod 4, and i | n}.
jvp> a(n) = PRODUCT{j == 1 mod 6, and j | n}* PRODUCT{k == 5 mod 6, and k | n}.
jvp> 
jvp> The first of these might be said to use the Gaussian core of n (the
jvp> product of Gaussian primes, A002145, of the form 4i+3 dividing n) and
jvp> the Pythagorean core of n (the product of Pythagorean primes, A002144,
jvp> of the form 4h+1 dividing n). Are these in OEIS?
jvp> ...


The first product mentioned is (for n=1,2,3,....)

Prod(h: h=1 mod4 and h|n):
1, 1, 1, 1, 5, 1, 1, 1, 9, 5, 1, 1, 13, 1, 5, 1, 17, 9, 1, 5, 21, 1, 1, 1, 125,
13, 9, 1, 29, 5, 1, 1, 33, 17, 5, 9, 37, 1, 13, 5
It is not actually a "core," because, as the example of a(25)=1*5*25=125 shows,
the product of these divisors may become larger than n.

The second product is
Prod(h: h=3 mod4 and h|n):
1, 1, 3, 1, 1, 3, 7, 1, 3, 1, 11, 3, 1, 7, 45, 1, 1, 3, 19, 1, 21, 11, 23, 3,
1, 1, 81, 7, 1, 45, 31, 1, 33, 1, 245, 3, 1, 19, 117, 1

The element-by-element product of these 2 sequences is
Prod(h: h=1 mod4 and h|n)*prod(i: i=3 mod 4 and i|n):
1, 1, 3, 1, 5, 3, 7, 1, 27, 5, 11, 3, 13, 7, 225, 1, 17, 27, 19, 5, 441, 11,
23, 3, 125, 13, 729, 7, 29, 225, 31, 1, 1089, 17, 1225, 27, 37, 19, 1521, 5

Prod(h: h=1 mod6 and h|n):
1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 13, 7, 1, 1, 1, 1, 19, 1, 7, 1, 1, 1, 25,
13, 1, 7, 1, 1, 31, 1, 1, 1, 7, 1, 37, 19, 13, 1

Prod(h: h=5 mod6 and h|n):
1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 11, 1, 1, 1, 5, 1, 17, 1, 1, 5, 1, 11, 23, 1, 5,
1, 1, 1, 29, 5, 1, 1, 11, 17, 175, 1, 1, 1, 1, 5

Prod(h: h=1 mod 6 and h|n)*prod(i: i=5 mod 6 and i|n):
1, 1, 1, 1, 5, 1, 7, 1, 1, 5, 11, 1, 13, 7, 5, 1, 17, 1, 19, 5, 7, 11, 23, 1,
125, 13, 1, 7, 29, 5, 31, 1, 11, 17, 1225, 1, 37, 19, 13, 5

Note that the products above have *not* been restricted to prime factors.
(see also A007955, product of all divisors of n)
--
Richard http://www.strw.leidenuniv.nl/~mathar