### Partner to the Prime Numbers: Highly Composite numbers, or, Quantitative Divisorial Initiation

Revised Sept 2013

On The Takeway (12/12/12), John Hockenberry noted that 12 has 'lots of factors or divisors. Well, it has the most factors of any smaller number, as has only multiples of 12 than has any number smaller than that. I call this 'quantitative divisorial initiation' or QDI for short. 12-and-its-QDI-multiples is the ONLY family of integers that has QDI (for any number which is NOT a multiple of twelve, it never has QDI: none of the numbers which are not multiple of twelve are QDI numbers, since some smaller number which IS a multiple of 12 initiated the quantity of divisors which that larger number merely replicates.

In other words, a number which has a greater quantity of divisors than any one of its predecessors is a number which thereby can be described as having 'initiated' the increase in quantity of divisors. No two QDI numbers are contiguous, but are rather separated by at least twelve other numbers. So, while in many cases the yard, or separating space, contains numbers which equal the quantity of divisors of the lesser QDI, no gap number exceeds the lesser QDI in terms of quantity of divisors. And, since the minimum gap is twelve numbers long, and since every QDI is a multiple of twelve, there is a lot of variation in quantity of divisors between any two 'next door' QDI numbers but never more than that of the lesser QDI.

3 and 6 are the only single integers with QFI. Algebraically speaking, 6 has two factors (i.e., 'distinguishing factors', since 1 and n are common to all numbers), and no number smaller than 12 has more that two distinguishing factors.
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#### But, other than these kinds of these finely tuned 'marching orders', the prime numbers also are extremely unpredictable: they occur or not occur without strict one-to-one predictability. Primes are like the 'Rambo of numbers.'

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FYI: I believe there is relative utility within any set of algorithms intended to more-or-less predict, or find more-or-less predictive patterns for, the occurrence of the prime numbers: not all such algorithms are equally useful to that intent. In other words, I believe that, given all the various attempts, to date, to predict the primes, there is a general narrowing down of the predictive power of algorithms so intended. But, I believe there logically can never be any sub-set of all algorithms-so-intended that can actually strictly predict the primes. Which means I believe there is no finite maximum of such algorithms that can perfectly predict the primes.
So, twelve and its multiples are numbers that 'mother the most children', for which the factors of those numbers are the children. This is reflected by 2 as the root QDI number over 1, and also as reflected by 6 as the highest of the smallest non-primes with a continuous string of factors:

#### 34 is divisible by 2, 1, and 17 (no better than for 6! AHH! A shark just brushed our leg!!!).

35 is divisible by 1, 5, and 7 (AAAHH! Another shark just brushed our leg!).

#### This reflects the root partnership between 2 and 6. So, beginning with 12, every sixth number-divisible-by-two is a QFI number.

QFI permits an exact 'grid' for the entire natural series, in terms of QFI=6 as multiples of twelve. Compare this to a 'grid' using any other multiple, say, multiples of 5 or 10: such an alternate 'grid' is a more arbitrary, less-interrelated, 'grid', partly because, unlike twelve, 10 lacks a continuous string of factors (1, 2, and 5 for 10; whereas 1, 2, 3, and 4 for 12). Any number with a continuous string of at least four factors is divisible by 12, but not any number with a continuous string of three factors is divisible by 10 or 5. The 'grid' to which I refer is keyed to those four continuous factors; So, for example, the sum of the factors of 12 is 10, and the sum of 10-and-12 is 22.