largest density of a prime signature

[David Corneth]

Hi all,

So I was wondering, if all numbers with a certain prime signature have
density 0 in the positive integers, what's the number with the largest
density so far?

By signature I mean the exponents with multiplicity, sorted.
So [1,2] and [2,1] are considered the same signature. [1,1,2] and
[1,2] are different signatures; multiplicity matters.

For signature [] (empty product) the only number having it is 1. The
largest density is 1.
For the signature [1] (the primes) the largest density is at 3 at
which we have a density of 2/3.
In the set S = {1,2,3} we have |S| = 3 and in S are two primes. So density 2/3.
There can never be a larger density as after that at most one of two
numbers are prime and 4 is not prime.

other prime powers with signature [m] have largest density at 2^m.

So I looked for other signatures record places for density of
signatures. From here on values are conjectured.
For [1,1] (Squarefree semiprimes, A006881) I get largest density at 95.
There are 30 numbers <= 95 with signature [1,1]. That's density 30/95 = 6/19.
Up to 100000 this record is not broken.

For signature [2,1] I get 76 with density 12/76 = 3/19.
For n = 1,10 I get largest density for signature [n, 1] at 95, 76, 56,
208, 224, 448, 896, 1792, 3584, 7168 respectively. The trailing bit
*seems* to be A005009, 7*2^n.

For [n, 2] I get 76, 36, 108, 400, 288, 576, 1152, 2304, 4608, 9216
which seems to be 9*2^n for large enough n.

[1,1,1] get quite large, largest density at 7109105. There are 1484160
numbers with signature [1,1,1] up to 7109105. The record 1484160 /
7109105 is not broken for any k in [7109106, 10^8]. Up to 7109105 the
record for largest density is broken 17484 times.

For [2,1,1] I get 12645 with density 1212/12645.

This way I could do a bunch more. But they're all conjectures.
For [1,1,1,1] I did a search but there's no slowing down in record breaking.

To find more for [1,1] we could look at the largest number of
squarefree semiprimes we can have in [95*m + 1, 95*m + 95] for m >= 0.
The record seems to be 34. Similar intervals for other signatures.

Could any of this be interesting for OEIS? I didn't find the number
7109105 index of record for [1,1,1] in OEIS.

Best,
David

[Sean A. Irvine]

On Sat, 25 Apr 2026 at 21:53, David Corneth <davida...@gmail.com> wrote:

Could any of this be interesting for OEIS? I didn't find the number
7109105 index of record for [1,1,1] in OEIS.

Certainly. Seems completely legitimate to me and interesting.