Separable combintations and strong divisibility

• Shiro Ando,
• Daihachiro Sato

Abstract

Given a positive integer sequence {a _n}, where n = 1,2,3,…, we replace n! in the defining form

(nr)=n!r!s!,  r+s=n

of the binomial coefficients with ∏an=anan−1⋯a2a1 , where we define ∏a0=1 . When the resulting numbers

[nr]=∏an∏ar∏as,  r+s=n

are all integers, we call them the generalized binomial coefficients defined by {a _n}, and the sequence {a _n} a Raney sequence. In particular, the Fibonacci sequence {F _n} is a Raney sequence, and the generalized binomial coefficients defined by it are called Fibonomial coefficients.

Find the separable sets of all combinations, and you have primes.

Another clue:

Alexander Bogomolny

For every integer n > 0 there exists an n-digit integer, with digits 1 and 2, divisible by 2ⁿ. 

Strong divisibility.