# [seqfan] Primorial numbers

Creighton Kenneth Dement creighton.k.dement at mail.uni-oldenburg.de
Mon Sep 21 23:15:17 CEST 2009

```Dear Seqfans,

I'm currently working on a full-time project which has nothing to do with
math, but a sequence came to me out of the blue and, under those
circumstances, I tend to stop what I'm doing...

The question that came to me was that since
prod(i=1, n, prime(i)) and
prod(i=1, n, prime(n+1)%prime(i))
are "surely" both core sequences, what then are the (reduced) numerators
and denominators when we divide the top sequence by the bottom sequence?

In fact, only the first sequence is a core sequence: A002110 (primorial
numbers). The second sequence, A102647 (apart from initial term), is
currently just a "plain sequence" with hardly a comment- which surprised
me a bit.

In all, I would like to submit 4 sequences: a1(n), b1(n), a2(n), b2(n)
(defined below)

nup(n)=prod(i=1, n, prime(i))
dep(n)=prod(i=1, n, prime(n+1)%prime(i))
nus(n)=sum(i=1, n, prime(i))
des(n)=sum(i=1, n, prime(n+1)%prime(i))

(PARI)
a1(n)=if(n<0, 0, numerator(nup(n)/dep(n)))
b1(n)=if(n<0, 0, denominator(nup(n)/dep(n)))
a2(n)=if(n<0, 0, numerator(nus(n)/des(n)))
b2(n)=if(n<0, 0, denominator(nus(n)/des(n)))

for(n=1, 10, print1(a1(n) ", "))
2, 3, 15, 105, 385, 5005, 17017, 323323, 1062347, 30808063,

for(n=1, 10, print1(b1(n) ", "))
1, 1, 1, 4, 6, 48, 64, 96, 576, 1728,

for(n=1, 10, print1(a2(n) ", "))
2, 5, 5, 17, 28, 41, 58, 77, 50, 129,

for(n=1, 10, print1(b2(n) ", "))
1, 3, 2, 8, 13, 18, 27, 29, 23, 56

Question: Do we always have a2(n)/b2(n) < 3
Printing only the last two results:
for(n=2998, 3000, print1(a2(n)/b2(n) +0.0 ", "))
2.764512455487481516352086719, 2.765020411098228343146854373,

Sincerely,
Creighton

```