Brilliant Numbers

8 is the smallest 3-brilliant number

Recently, my day count returned a pair of brilliant numbers: 24287 and 24289. A brilliant number is defined by Peter Wallrodt as a number with two prime factors of the same length (in decimal notation). These numbers are generally used for cryptographic purposes, and for testing the performance of prime factoring programs. It is possible to define n-brilliant numbers as the product of n prime numbers of the same length (source).

By this definition, 24287 = 149 x 163 and 24289 = 107 x 227 are 2-brilliant numbers. What's interesting about these two brilliant numbers is that they are consecutive (because there are no even brilliant numbers greater than 14). As is explained in OEIS A083284:

The only consecutive brilliant numbers are {9, 10} and {14, 15}; and for n > 14 there are no brilliant constellations of the form {n, n+(2k+1)} or equivalently {n, 2k+n+1} with k >= 0. Proof: One of n and 2k+n+1 will be even. And there are no even brilliant numbers > 14 since they must have the form 2*p where p is a prime having only one digit.

Here is the list of the smaller of the brilliant numbers pairs (from OEIS A083284):

Numbers n such that n and n+2 are both brilliant numbers, where brilliant numbers are semiprimes whose prime factors have an equal number of decimal digits, or whose prime factors are equal.

4, 527, 779, 869, 899, 1079, 1157, 1271, 1679, 4187, 6497, 6887, 24287, 24881, 25019, 29591, 35237, 37127, 37769, 38807, 39269, 39911, 41309, 43361, 44831, 45347, 46001, 46127, 47261, 48509, 48929, 51809, 52907, 54389, 55481, 55751, 55961