Semiprime Triples

I track the properties of the number associated with my diurnal age using Airtable and today I noticed an interesting pattern. See Figure 1.

Figure 1

Referring to Figure 1, it can be seen that I'm experiencing a run of three consecutive semi-primes. I thought I'd investigate how frequent such runs were. I'm regarding a semi-prime as a number that is the product of two distinct prime numbers and thus excluding square numbers like 121. It should also be pointed at that a run of four consecutive semi-primes is not possible. OEIS A039833 uses the smallest of the semi-primes to mark such patterns:

A039833

Smallest of three consecutive square-free numbers \(k, k+1, k+2\) of the form \(p \times q\) where \(p\) and \(q\) are primes.

Up to 1000, there are 13 such triplets with the first being (33, 34, 35):

• (33, 34, 35)
• (85, 86, 87)
• (93, 94, 95)
• (141, 142, 143)
• (201, 202, 203)
• (213, 214, 215)
• (217, 218, 219)
• (301, 302, 303)
• (393, 394, 395)
• (445, 446, 447)
• (633, 634, 635)
• (697, 698, 699)
• (921, 922, 923)

This gives a frequency of 1.3%. Tracking the frequency in powers of 10 (starting with 100) produces the following results: 

• 2 in the first 100, a percentage of 2%
• 13 in the first 1,000, a percentage of 1.3%
• 71 in the first 10,000, a percentage of 0.71%
• 379 in the first 100,000, a percentage of 0.379%
• 2377 in the first 1,000,000, a percentage of 0.2377%

This is shown in Figure 2 using a log scale for the horizontal axis.

Figure 2: permalink

These results are not surprising because as the numbers get larger, it is less likely that they'll have only two distinct prime factors. It would seem that, asymptotically, the frequency approaches zero.

The next such triple is not far off, being (26581, 26582, 26583). It should be pointed out that all semi-prime triples must have one number with 2 as a factor and another number with 3 as a factor. These two numbers must be consecutive in either order. The third number, either the smallest or largest of the triple, will be a semi-prime that does not have 2 or 3 as a factor. Let's use this triple as an example:

• 26581 = 19 x 1399
• 26582 = 2 x 13291
• 26583 = 3 x 8861

Mention should be made of the situation where two semi-prime triples are separated by a single number (which must always be divisible by 36). The first such pair of triples is (213, 214, 215) and (217, 218, 219) where the dividing number 216 = 6 x 36. OEIS A202319 uses the middle semi-prime in the first member of the pair to mark such patterns (thus 214 is the first member):

A202319

Lesser of two semi-primes sandwiched each between semi-primes thus forming a twin semi-prime triple.

The initial members of the sequence are:

214, 143098, 194758, 206134, 273418, 684898, 807658, 1373938, 1391758, 1516534, 1591594, 1610998, 1774798, 1882978, 1891762, 2046454, 2051494, 2163418, 2163958, 2338054, 2359978, 2522518, 2913838, 3108202, 4221754, 4297318, 4334938, 4866118, 4988878, 5108794