Another Look At Semiprimes

 I've written about semiprimes before, specifically in the following posts:

• Brilliant Numbers
• Pandigital Numbers Formed From the Product of a Number and its Reversal
• Golden Semiprimes
• An Unhappy Family
• A Prime to Remember
• 2019: A Numerical Profile
• Semiprime Factor Ratios
• More about Golden Semiprimes
• 2019: A Numerical Profile

SEMIPRIME TRIPLETS

However, as with most mathematical topics, there's always more to discover. Today I turned \(26583\) days old and one of the properties of the number \(26583\) is that it's a member of OEIS A115393:

A115393

Numbers \(n\) such that \(n\), \(n-1\) and \(n-2\) are semiprimes.

So we find that:

• \(26583 = 3 \times 8861\)
• \(26582  = 2 \times 13291\)
• \(26581 = 19 \times 1399\)

It's not possible to have four semiprimes in a row because every fourth number must be a multiple of \(4\). We see that here because:

• \(26584 =2^3 \times 3323\)
• \(26580 = 2^2 \times  3 \times  5 \times 443\)

In the range from \(1\) up to \(26583\) there are \(139\) such triplets. The sequence begins:

35, 87, 95, 123, 143, 203, 215, 219, 303, 395, 447, 635, 699, 843, 923, 1043, 1139, 1263, 1347, 1403, 1643, 1763, 1839, 1895, 1943, 1983, 2103, 2183, 2219, 2307, 2363, 2435, 2463, 2519, 2643, 2723, 2735, 3099, 3387, 3603, 3695, 3867, 3903, 3959, 4287

The first triplet is thus:

• \(33=3 \times 11\)
• \(34=2 \times 17\)
• \(35=5 \times 7\) 
  
  

RECORD RUNS OF NUMBERS THAT ARE NOT SEMIPRIMES

What about record runs of numbers that are not semiprimes? It turns out that \(6252893229398\) marks the start of a record-breaking run of \(173\) consecutive integers that ends with \(6252893229570\). The second case of a run of the same length is between \(9189221611478\) and \(9189221611650\). There are no greater runs less than \(10^{13}\). Source. 

These numbers and their factorisations can be viewed by following this permalink. The semiprimes before and after the first record-breaking run are:

• \(6252893229397 = 83537 \times 74851781\)
• \(6252893229571 = 609607 \times 10257253\)

For the second record-breaking run, the semiprimes before and after are:

• \(9189221611477 = 877 \times 10478017801\)
• \(9189221611651 = 197 \times 46645794983\)

The numbers in between, together with their factorisations, can be viewed by following this permalink.

THE ARECIBO MESSAGE

Figure 1: This is a demonstration of the message with
colour added to highlight its separate parts.
The binary transmission sent carried no colour information.

An interesting use of semiprimes is the Arecibo message involving the use of the semiprime \(1679\). See Figure 1.

The number \(1679\) was chosen because it is a semiprime (the product of two prime numbers), to be arranged rectangularly as \(73\) rows by \(23\) columns. The alternative arrangement, \(23\) rows by \(73\) columns, produces an unintelligible set of characters.

SEMIPRIME COUNTING FORMULA

A semiprime counting formula was discovered by E. Noel and G. Panos in 2005. Source: On distribution of semiprime numbers: Shamil Ishmukhametov.

Let \( \pi_2 (n) \) denote the number of semiprimes less than or equal to \(n\).  Then$$ \pi_2 (n) = \sum_{k=1}^{\pi (\sqrt n) } [ \pi(n/p_k) - k + 1 ]$$where \( \pi(x) \) is the prime-counting function and \(p_k\) denotes the \(k\)th prime. Source: Semiprime from Wolfram MathWorld.

This formula does return, correctly, the result that 26583 is the 6648th semiprime (permalink).