[seqfan] Distance to the nearest prime in numbers that are products of some number of primorial numbers?

[Antti Karttunen]

Cheers all,

consider sequences like Highly Composite Numbers, A002182, or
https://oeis.org/A129912 "Numbers that are products of distinct
primorial numbers". It seems that in these sequences the nearest prime
to any of their terms larger than 2, in both directions, is always
either adjacent or a prime distance away from that term, like is observed
by Bill McEachen in A117825.

However, for A025487, which consists of all products of primorials,
thus is a supersequence of both aforementioned sequences, this is not
always true: 512 is a member of A025487, but the next prime after 512
is 521, with 521 - 512 = 9, and nine is a composite number.

This seems to be a more general form of Fortune's conjecture (which
applies only to primorials themselves, and only to primes >  them).
See https://oeis.org/A005235
and also https://oeis.org/A060270 (for distances to primes < primorials).

I can only see so far that the distance h of the nearest prime p from
such a primorial product x, if it is not 1, but instead either a prime
or composite, must have the smallest prime factor which is greater
than the greatest prime factor of x, because we must have gcd(p,x) =
1, for p is a prime not present in the product x of primorials, thus
as p = x+h then gcd(x+h, x) = gcd(x, h)  = 1 as well [also when p =
x-h, we have 1 = gcd(x-h,x) = gcd(x,h)], thus h cannot share a prime
factor with x either (and x being a primorial product, all the primes
from 2 up to the largest prime of x are no more available).


Best regards,

Antti

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