Primitive Practical Numbers

On July 31st 2018, three years and one day ago, I created a post titled Practical Numbers. To quote from that post:

In number theory, a practical number or panarithmic number is a positive integer \(n\) such that all smaller positive integers can be represented as sums of distinct divisors of \(n\). For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2. Wikipedia.

That is straightforward enough and, before we progress to the concept of primitive practical numbers, let's familiarise ourselves with the concept of a complete sequence. Here is a definition:

In mathematics, a sequence of natural numbers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.

For example, the sequence of powers of two (1, 2, 4, 8, ...), the basis of the binary numeral system, is a complete sequence; given any natural number, we can choose the values corresponding to the 1 bits in its binary representation and sum them to obtain that number (e.g. \(37 = 100101_2 = 1 + 4 + 32\)). This sequence is minimal, since no value can be removed from it without making some natural numbers impossible to represent. Simple examples of sequences that are not complete include the even numbers, since adding even numbers produces only even numbers—no odd number can be formed. Wikipedia.

Primitive practical numbers are defined in OEIS A267124:

A267124

Primitive practical numbers: practical numbers that are square-free or practical numbers that when divided by any of its prime factors whose factorisation exponent is greater than 1 is no longer practical.

 The accompanying comments in the OEIS entry are informative:

If \(n\) is a practical number and \(d\) is any of its divisors then \(n \times d\) must be practical. Consequently, the sequence of all practical numbers must contain members that are either square-free (A265501) or when divided by any of its prime factors whose factorisation exponent is greater than 1 is no longer practical. Such practical numbers are said to be primitive. The set of all practical numbers can be generated from the set of primitive practical numbers by multiplying these primitives by any of their divisors.

The examples are given of:

• \(a(4)=20=2^2 \times 5\). It is a practical number because it has 6 divisors 1, 2, 4, 5, 10, 20 that form a complete sequence. If it is divided by 2 the resultant has 4 divisors 1, 2, 5, 10 that is not a complete sequence. Thus it is a primitive practical number.
  
  
• \(a(7)=42=2 \times 3 \times 7\). It is square-free and is practical because it has 8 divisors 1, 2, 3, 6, 7, 14, 21, 42 that form a complete sequence. Thus it is a primitive practical number.

12 as an example of a practical number that is not primitive. It has divisors 1, 2, 3, 4, 6, 12 that form a complete sequence. It is not square-free and, when divided by 2 (a prime factor whose factorisation exponent is greater than 1), the result is 6 which is still a practical number. 6 in fact is a primitive practical number because it is a practical number that is square-free.

The initial primitive practical numbers are:

1, 2, 6, 20, 28, 30, 42, 66, 78, 88, 104, 140, 204, 210, 220, 228, 260, 272, 276, 304, 306, 308, 330, 340, 342, 348, 364, 368, 380, 390, 414, 460, 462, 464, 476, 496, 510, 522, 532, 546, 558, 570, 580, 620, 644, 666, 690, 714, 740, 744, 798, 812, 820, 858, 860, 868, 870, 888, 930, 966, 984 

Let's examine the first 8 primitive practical numbers: 1, 2, 6, 20, 28, 30, 42 and 66. The differences between these terms are 1, 4, 14, 8, 2, 12 and 24. The record gaps are 1, 4, 14 and 24, which occur after the terms 1, 2, 6 and 42. If we extend our examination of these record gaps and note the terms before which they occur, we end up with OEIS A334883:

A334883

Primitive practical numbers (A267124) with a record gap to the next primitive practical number.

The initial terms making up this sequence are:

1, 2, 6, 42, 104, 140, 1036, 1590, 2730, 7900, 10374, 19180, 22660, 23180, 26418, 105868, 114960, 139060, 295780, 403524, 482250, 1294144, 2468944, 4799058, 5379282, 19035500, 20233936, 21803860, 112406992, 789190976, 3520928922

What drew my attention to this sequence and to primitive practical numbers in general was the fact that my diurnal age today is 26418. The gap to the next primitive practical number 26572 is 154 but that record is not broken until 105868, after which there is a gap of 188 to 106056.