cp's OEIS Frontend

It is conjectured that for all integers m there exist two primes p and r such that all the binomial coefficients (m,k) with 1 <= k <= m-1 are divisible by either p or r. Using Lucas's Theorem we can prove that the conjecture is true for integers m such that the difference between m and the largest prime smaller than m is smaller than the greatest prime power divisor of m. In A290203 we examine the integers that do not satisfy this property. However, when this property is not satisfied, if the difference between m and twice the largest prime smaller than m/2 is smaller than the greatest prime power divisor of m we can prove the conjecture for odd numbers and some cases of even numbers. Note that we take the difference between m and twice the largest prime smaller than m/2 because this is the last digit of the base-p representation of m, where p refers to the largest prime smaller than m/2. Therefore, this list examines the numbers that satisfy neither of the two properties and hence are the difficult cases for the proof of the conjecture.