**What is the opposite of a prime number?**

Every now and then I skip my lunch break and strap myself onto a gym’s rowing thwart to employ the meal time instead for a counterbalance of desk work and body mass index. For that I have about half an hour of spare time – i.e. some 30 minutes or 1800 seconds, respectively. The time set is directly eligible at the rowing machine and counts down the remaining time to zero.

During the sport exercise I adopted the habit of an accompanying mental exercise to calculate the moment when the remaining duration becomes an integer fraction of the time set. For example, starting with 20 minutes (‘) at remaining 10’ means the half to go, at 6’40 means a third, at 5’ a quarter, at 4’ a fifth, at 3’20 a sixth, at 2’30 an eighth, a 2’ a tenth and so on. As is known, the divisors succeed by ever decreasing intervals when approaching the end – which corresponds quite pleasantly with my exhaustion. For, the lower the power reserves get, the faster one gets to the next time gate.

However, it is somewhat annoying that some divisors do not fit in completely. Within the example given it is the seventh and the ninth, which do not fit in the flow of seconds without remainder. Therefore, I have increased the time set to 21’, which then allows a half at 10’30, a third at 7’, a quarter at 5’15, a fifth at 4’20, a sixth at 3’30, a seventh at 3’, a ninth at 2’10 and a tenth at 2’06. This gives a better matching of the count-down – however, now sacrificing the eighth.

Which time set has to be chosen to include the eighth? And which time set would be anyhow the best to include as much divisors as possible during a reasonable break?

**LCM**

For sure, you may employ the lowest common multiple LCM to get different fractions to a common denominator. Therefore, you split up all numbers into prime numbers and form the product of the highest available prime powers. For the numbers from 1 to 12 that would give 2^{3}×3^{2}×5×7×11, however, the result is 27,720 seconds or 462 minutes, i.e. almost 8 hours. You probably agree that it is preferable to look for a number below a preset limit – say 1800 seconds – which provides as much divisors as possible. Yet, what kind of numbers are these?

To begin, 1 has one divisor, 2 has two of them, namely 1 and 2. But 3 has also two just divisors, before 4 raises to three of them, namely 1, 2 and 4. Next it is the 6, which raises to four divisors (1; 2; 3; 6). Afterwards one has to wait until 12, yet already with a jump to six divisors (1; 2; 3; 4; 6; 12). And then it lasts again twice before 24 increases again by 2 devisors to 8 (1; 2; 3; 4; 6; 8; 12; 24). However, this is not a rule, because the next step provides just one devisor more, yet occurs already at 36 (1; 2; 3; 4; 6; 9; 12; 36). Further, there will by 48 with 10 devisors and 60 with 12. And, then again, there is a big gap: Only at 120 you will obtain more devisors, although instantly 16 of them.

A convenient side-effect of these considerations has been for me, that time flew by and the physical stress dropped behind the mental one. In fact, the physical power indicated on the ergometer had become higher than before, because the problem had obviously enhanced my fervor.

The abstract formulation of the task is:

**· ** Which number in a natural sequence provides more devisors than any others before?

**Deprime or HCN**

At first, I have baptized them “deprime numbers”, because they seem to be somewhat like the opposite of prime numbers. However, I had to learn that they are already known in mathematics as high composite numbers HCN – and already the outstanding mathematical genius Srinivasan Ramanujan (1887-1920) has worked on them. Furthermore,

the founders of historic unit systems seem to have understood the relevance of HCN, because a day is divided into 24 hours and an hour into 60 minutes and a whole circle into 360 degrees – all of them being deprime numbers. Most interestingly, even natural systems possess sometimes a corresponding plurality, for example the (almost) 12 lunar cycles within a solar cycle during (approximately) 360 days. Somehow, the HCNs appear to be as interesting as prime numbers.

The connection between prime and deprime numbers is given by the usual prime factorization available for all natural numbers: n = Πp_{i}^{ci}

If n is a prime number, then there are just two combinations i of prime factors p, i.e. 1 and n. All other numbers offer more opportunities for divisor combinations d with different prime numbers, which can be calculated as: d = Π(c_{i }+ 1).

In this context, a HCN is that natural number n, which corresponds to a number of divisors d that is higher than all others before. Since the double of a HCN provides at least more combinations of prime factors, the number of divisors can rise without limits. Although the value of the deprime numbers HCN increases quite fast, there are infinite numbers of them as well.

**HDN**

Inversely, each HCN corresponds exactly to just one high divisor number HDN. And one may assume that each natural number corresponds to a HDN and a corresponding HCN. However, although there are equally infinite HDN, there is not an HDN for each natural number, because already in the first elements of the sequence – exemplified above – the 5 and 7 are missing as HDN. Hence, the sets of HCN and HDN represent two corresponding infinite subsets of the natural number.

[HCN] = (1, 2, 4, 6, 12, 24, 36, 48, 60, 120, …)

[HDN] = (1, 2, 3, 4, 6, 8, 9, 10, 12, 16, …)

What would be the right answer in an IQ-Test about to continue the number sequences?

What would it signify in regard to popular numerology of “The Da Vinci Code” and others?

The paired numbers (HCN; HDN) increases monotonically, because each bigger number of divisors corresponds to a bigger deprime number.

But what about the relation r = HCN/HDN.

Within the first values of r, it decreases monotonically: 1/1; 2/2; 3/4; 4/6; 6/12; 8/24; 9/36; 10;48; 12/60; 16/120.

But, does this sequence converge to zero – or asymptotically to another value? Are there any regularities or rules to derive HCN, HDN or their relation, at least? Are there structural similarities to prime numbers?

Intuitively, you may believe that HDN will occur more often for higher HCN, since the prime numbers get more and more pinched out – which may raise the possible combinations for prime power.

Anyhow, these numbers seem to hide some secrets …

**Utilities**

How many delegates should have an eligible committee in order to represent as much fractions as possible.

Each election and ballot has this problem with appropriate representation. Since each deputy requires resources – by remuneration, publicity and parliamentary debate and opinion formation – it may be advisable to adjust the available seats to 121 with 3 possible fractions, in spite of 120 with 16 possible fractions … although a further deputy is to supply.

How should a regular meeting be fixed in order to harmonize with as much other periodic events as possible?

In fact, the weekly period of 6 working days – the monthly of 30 days or the yearly of 12 months – seems suitable to harmonize with each other.

Once again, it appears to contrast the known prime number phenomenon of “periodical cicadas”, which remain for a prime number of years hidden in the underground. With 7, 13 or even 17 years of development, it is suggested that they outwait their predators, because they never suit to a multiplicity of minor numbers. Perhaps, a closer look may reveal a species with a deprime number cycle of 6, 12, 24 or even 60 months or years in order to share efficiently the occasions for partnering or prey.

How many parts should contain a stock in order to get split up into as many fractions as possible.

“Lean thinking” in modern industrial production has the principle of a fixed “takt time” to combine a smooth workflow with utmost flexibility. If the components for an assembly vary – such as screws, nuts, shims or bars –, or the emergency supply has to be occasionally adjusted to the number of persons and days, an orientation by HCN or

HDN may help to improve production or emergency supply – and prohibit nonproductive remnants, expensive subsequent deliveries or distribution riots.

Somewhat “romantic” is the coincidence that the number 2 is both prime as well as deprime. It is the origin of oppositional infinite sequences, i.e. the “lone” prime numbers and the “social” deprime numbers. This sounds somewhat compliant with usual problems in ambiguities: To get an individual part of a pair – and to stay self dependent as a

couple.

Somewhat “romantic” is the coincidence that the number 2 is both prime as well as deprime. It is the origin of oppositional infinite sequences, i.e. the “lone” prime numbers and the “social” deprime numbers. This sounds somewhat compliant with usual problems in ambiguities: To get an individual part of a pair – and to stay self dependent as a

couple.

This basic difference has some relevance for theoretical quantum mechanics: While fermions bear on the Pauli principle of “lonely” exclusion – i.e. two quantum objects with identical quantum numbers at the same place are completely forbidden – bosons bear on the principle of “social” inclusion for a Bose-Einstein condensate – i.e. as many

quantum objects with identical quantum numbers at the same place is highly probably. Maybe the analogy with prime numbers and HCN provides a novel opportunity for a mathematical description, e.g. for quantum organization of atoms or lasers.

In contrast, the practical impact for my lunch break on a rowing thwart appears quite low. Although I meanwhile figured out that 1680 seconds (28’) provides 40 divisors and my first approach with 21’ (1260 seconds) with 36 divisors has already been quite good. Yet, I suspect that there are many other problems of practical life, where a smooth division is helpful to increase the efficiency

When I used to live in Paris, there were bus services on the periphery highway, which just service each second, third or particular stations in order to increase the mean velocity of the schedule speed. It may turn out advantageous to implement a deprime number of stations, which are serviced by busses with many different intervals.

[However, according to the perpetual road blockages there has never been something like a regular service, which could be improved. If it was indicated that a regular service was provided each 5 minutes, it just meant that every half an hour 6 busses came on at once …]