Numbers Like Fourteen

The number \( 14 \) has the property that the sum of its digits is equal to the difference of its prime factors. That is, \( 14 = 2 * 7 \) and \( 1 + 4 = 7 - 2 \).

Let’s say that a number is like fourteen if it is the product of two primes and its digit sum is equal to the difference of its prime factors. The first fifty numbers like fourteen are:

14, 95, 527, 851, 1247, 3551, 4307, 8051, 14351, 26969, 30227, 37769, 64769, 87953, 152051, 163769, 199553, 202451, 256793, 275369, 341969, 455369, 1070969, 1095953, 1159673, 1232051, 1625369, 1702769, 2005007, 2081993, 2116769, 3674633, 4040051, 4397153, 4523873, 4600769, 4941473, 5075753, 5405369, 5630873, 6036593, 6593999, 7144673, 7305953, 7935233, 9941153, 9997619, 10304051, 10477913, 11390369

The first 1000 numbers like fourteen can be downloaded here. This list and the observations below were made with variations of the python scripts found here.

Update 2019-08-03: OEIS sequence A308821.

Observations:

1. Given that \( 2 \) is the only even prime and \( p - 2 \) grows much faster than the digit sum of \( 2p \), it is clear that \( 14 \) is the only even number like fourteen.

2. Other than \( 14 \), every number like fourteen that I have found has a digit sum of \( 18n - 4 \) for some natural number \( n \). (Is this true for all odd numbers like fourteen?)

   digit sum	product	factor a	factor b
   14	95	5	19
   32	26969	149	181
   50	6593999	2543	2593
   68	399798869	19961	20029
   86	169987989767	412253	412339
   104	57779776889897	7601249	7601353
   122	1599996799997879	39999899	40000021

3. It is possible for multiple numbers like fourteen to share a factor. That is, there are prime numbers \( a < b < c \) such that either \( ab \) and \( ac \) are both like fourteen or \( ac \) and \( bc \) are both like fourteen. Here is an example:

   product	factor a	factor b
   25449182159	159503	159553
   25452054113	159521	159553
   25454925491	159521	159571
   25457796869	159521	159589
   25460669471	159539	159589

4. There are numbers like fourteen with nested factors. That is, there are prime numbers \( a < b < c < d \) such that \( ad \) and \( bc \) are both like fourteen. Here is one example:

   product	factor a	factor b
   199553	431	463
   202451	443	457

5. I haven’t found any prime numbers \( x < y < z \) such that \( xy \) and \( yz \) are both like fourteen. (Do such numbers exist?)

6. For every odd number like fourteen that I have found, the sum of the factors is a multiple of \( 6 \). If we write the factors as \( a < b \), then combined with observation 2 above, \( a = 6m - 1 \) and \( b = 6n + 1 \) for some \( m,n \in \mathbb{N} \). If this is true for all odd numbers like fourteen, that would mean that the answer to the question in observation 5 is no.

7. Not every combination of primes fitting the above patterns generates a number like fourteen. For example, \( 2867 = 47 * 61 \), and \( 2867 \) is not like fourteen.

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last updated: 2019-06-28