For the range 1 to 2310 for n:
additional proportions: (check these for larger primorials)
16/77 numbers have zero prime factors from 2,3,5,7,11:
00000 occurs 480/2310=16/77, so 16/77 of numbers don't have any of
prime factors 2 or 3 or 5 or 7 or 11.
44/105=16/77+8/77+4/77+8/231+8/385 numbers have one prime factor
from 2,3,5,7,11:
10000 occurs 480/2310=16/77, so 16/77 of numbers have prime factor 2
and don't have any of prime factors 3 or 5 or 7 or 11.
01000 occurs 240/2310=8/77, so 8/77 of numbers have prime factor 3
and don't have any of prime factors 2 or 5 or 7 or 11.
00100 occurs 120/2310=4/77, so 4/77 of numbers have prime factor 5
and don't have any of prime factors 2 or 3 or 7 or 11.
00010 occurs 80/2310=8/231, so 8/231 of numbers have prime factor 7
and don't have any of prime factors 2 or 3 or 5 or 11.
00001 occurs 48/2310=8/385, so 8/385 of numbers have prime factor 11
and don't have any of prime factors 2 or 3 or 5 or 7.
326/1155=8/77+4/77+8/231+2/77+8/385+4/231+4/385+2/231+2/385+4/1155
numbers have two prime factors from 2,3,5,7,11:
11000 occurs 240/2310=8/77, so 8/77 of numbers have prime factor 2
and 3 and don't have any of prime factors 5 or 7 or 11.
10100 occurs 120/2310=4/77, so 4/77 of numbers have prime factor 2
and 5 and don't have any of prime factors 3 or 7 or 11.
10010 occurs 80/2310=8/231, so 8/231 of numbers have prime factor 2
and 7 and don't have any of prime factors 3 or 5 or 11.
01100 occurs 60/2310=2/77, so 2/77 of numbers have prime factor 3
and 5 and don't have any of prime factors 2 or 7 or 11.
10001 occurs 48/2310=8/385, so 8/385 of numbers have prime factor 2
and 11 and don't have any of prime factors 3 or 5 or 7.
01010 occurs 40/2310=4/231, so 4/231 of numbers have prime factor 3
and 7 and don't have any of prime factors 2 or 5 or 11.
01001 occurs 24/2310=4/385, so 4/385 of numbers have prime factor 3
and 11 and don't have any of prime factors 2 or 5 or 7.
00110 occurs 20/2310=2/231, so 2/231 of numbers have prime factor 5
and 7 and don't have any of prime factors 2 or 3 or 11.
00101 occurs 12/2310=2/385, so 2/385 of numbers have prime factor 5
and 11 and don't have any of prime factors 2 or 3 or 7.
00011 occurs 8/2310=4/1155, so 4/1155 of numbers have prime factor 7
and 11 and don't have any of prime factors 2 or 3 or 5.
31/385=2/77+4/231+4/385+2/231+2/385+1/231+4/1155+1/385+2/1155+1/1155
numbers have three prime factors from 2,3,5,7,11:
11100 occurs 60/2310=2/77, so 2/77 of numbers have prime factor 2
and 3 and 5 and don't have any of prime factors 7 or 11.
11010 occurs 40/2310=4/231, so 4/231 of numbers have prime factor 2
and 3 and 7 and don't have any of prime factors 5 or 11.
11001 occurs 24/2310=4/385, so 4/385 of numbers have prime factor 2
and 3 and 11 and don't have any of prime factors 5 or 7.
10110 occurs 20/2310=2/231, so 2/231 of numbers have prime factor 2
and 5 and 7 and don't have any of prime factors 3 or 11.
10101 occurs 12/2310=2/385, so 2/385 of numbers have prime factor 2
and 5 and 11 and don't have any of prime factors 3 or 7.
01110 occurs 10/2310=1/231, so 1/231 of numbers have prime factor 3
and 5 and 7 and don't have any of prime factors 2 or 11.
10011 occurs 8/2310=4/1155, so 4/1155 of numbers have prime factor 2
and 7 and 11 and don't have any of prime factors 3 or 5.
01101 occurs 6/2310=1/385, so 1/385 of numbers have prime factor 3
and 5 and 11 and don't have any of prime factors 2 or 7.
01011 occurs 4/2310=2/1155, so 2/1155 of numbers have prime factor 3
and 7 and 11 and don't have any of prime factors 2 or 5.
00111 occurs 2/2310=1/1155, so 1/1155 of numbers have prime factor 5
and 7 and 11 and don't have any of prime factors 2 or 3.
23/2310=1/231+1/385+2/1155+1/1155+1/2310 numbers have four prime
factors from 2,3,5,7,11:
11110 occurs 10/2310=1/231, so 1/231 of numbers have prime factor 2
and 3 and 5 and 7 and don't have prime factor 11.
11101 occurs 6/2310=1/385, so 1/385 of numbers have prime factor 2
and 3 and 5 and 11 and don't have prime factor 7.
11011 occurs 4/2310=2/1155, so 2/1155 of numbers have prime factor 2
and 3 and 7 and 11 and don't have prime factor 5.
10111 occurs 2/2310=1/1155, so 1/1155 of numbers have prime factor 2
and 5 and 7 and 11 and don't have prime factor 3.
01111 occurs 1/2310=1/2310, so 1/2310 of numbers have prime factor 3
and 5 and 7 and 11 and don't have prime factor 2.
1/2310 numbers have five prime factors from 2,3,5,7,11:
11111 occurs 1/2310=1/2310, so 1/2310 of numbers have prime factor 2
and 3 and 5 and 7 and 11.
16/77+44/105+326/1155+31/385+23/2310+1/2310=1
Accumulate[{16/77, 44/105, 326/1155, 31/385, 23/2310, 1/2310}]
(* output: 16/77,724/1155,10/11,381/385,2309/2310,1 *)
A024451/A002110
2927/2310=(16/77)*0 + (44/105)*1 + (326/1155)*2 + (31/385)*3 +
(23/2310)*4 + (1/2310)*5 = 1/2 + 1/3 + 1/5 + 1/7 + 1/11
without reduced fractions:
2927/2310 == (480/2310)*0 + (968/2310)*1 + (652/2310)*2 +
(186/2310)*3 + (23/2310)*4 + (1/2310)*5 = 1/2 + 1/3 + 1/5 + 1/7 +
1/11
This gives the average number of distinct prime factors <=
prime(n)=11 in the natural numbers up to 2*3*5*7*11=2310
Mathematica
code for some other abcdef solutions