Human Genome Numbers

For want of a better term, I'll refer to the numbers that this post is about as Human Genome Numbers because the human genome is comprised of 46 chromosomes, 23 each of which are supplied by the father and mother. But first some background. I very much enjoy coming across numbers that mark my diurnal age and that have, seemingly, no interesting properties that are immediately discoverable. Such is the case today as I turn 27446 days old. My usual resources turned up nothing. 

Looking further however, I noted that the sum of this number's digits is 23, a very interesting number in its own right. I then noticed the number 46 formed by the last two digits of the number. Now 27446 factorises as follows:

$$27446 = 2 \times 13723$$ It can be seen that the 46 arises from the doubling of the last two digits in the factor 13723. This led me to seek out semiprimes with digit sums of 23 and with a smallest factor of 2 and a larger factor with 23 as its last two digits.

This turns out to be a rather exclusive club and there are only six such numbers in the range up to 40,000 (the top of the range that I normally work with). Here are the numbers and their factorisations:

$$\begin{align} 7646 &= 2 \times 3823 \\ 9446 &= 2 \times 4723 \\23846 &= 2 \times 11923 \\ 25646 &= 2 \times 12823 \\ 27446 &= 2 \times 13723 \\ 36446 &= 2 \times 18223 \end{align}$$ So to focus on 27446 is particular, we see that: $$\overbrace{27446}^{digits \, sum \, to \, 23} = 2  \times 13723 \text{ where 2 x 23 = 46}$$ There are 82 such numbers in the range up to one million and they are:

7646, 9446, 23846, 25646, 27446, 36446, 47246, 50846, 56246, 58046, 63446, 70646, 74246, 106646, 113846, 124646, 128246, 135446, 140846, 144446, 162446, 166046, 175046, 207446, 209246, 214646, 216446, 221846, 227246, 234446, 241646, 252446, 254246, 265046, 292046, 302846, 313646, 320846, 322646, 328046, 333446, 335246, 337046, 346046, 351446, 362246, 403646, 410846, 421646, 423446, 427046, 443246, 450446, 452246, 490046, 500846, 502646, 504446, 513446, 522446, 526046, 535046, 540446, 551246, 562046, 580046, 601646, 630446, 652046, 702446, 704246, 711446, 720446, 751046, 801446, 812246, 814046, 821246, 832046, 900446, 913046, 931046

There are other ways to search for the presence of 23 and 46 in numbers but the method I've chosen identifies 27446 and that was the intent. Let's repeat the criteria for selection:

• number must have a digit sum of 23
• number must be a semiprime with 2 as the smaller factor
• larger factor must end in the digits 23 (ensuring number itself ends in 46)

Two of the previous numbers have the digit sequence 23 occurring twice in the larger factor. These numbers are:

• 47246 = 2 x 23623
• 124646 = 2 x 62323 (notice that the number itself contains two instances of 46)

One number has the digit sequence 23 and 46 in the larger factor. This number is:

• 292046 = 2 x 146023 (notice that 46 also appears in the larger factor)

Another method of selection is to choose numbers with digit sums of 23 and with 46 = 2 x 23 as a factor. This selection method yields 47 numbers in the range up to 40,000. These numbers are:

1886, 3956, 4784, 5198, 6854, 7268, 7682, 8096, 8924, 9338, 9752, 10994, 13478, 13892, 15548, 15962, 16376, 16790, 17618, 18446, 18860, 19274, 21758, 22586, 23828, 24656, 25484, 26726, 27554, 28382, 29624, 30866, 31694, 32936, 33764, 34178, 34592, 35834, 36248, 36662, 37076, 37490, 37904, 38318, 38732, 39146, 39560

Take the first member in the sequence as an example:

• 1886 = 2 x 23 x 41 = 46 x 41

The common thread that justifies the selection of what might be termed human genome numbers is that the numbers 23 and 46 must be involved in some way. This opens the door for other selection criteria but I'll leave off at this point and maybe return to this theme at a later date.