Adams theorem on Bernoulli numbers revisited

If we denote Bn to be nth Bernoulli number, then the classical result of Adams (J. Reine Angew. Math. 85 (1878) 269) says that pℓn and (p−1) n, then pℓBn where p is any odd prime p>3. We conjecture that if (p−1) n, pℓn and pℓ+1 n for any odd prime p>3, then the exact power of p dividing Bn is either ℓ or ℓ+1. The main purpose of this article is to prove that this conjecture is equivalent to two other unproven hypotheses involving Bernoulli numbers and to provide a positive answer to this conjecture for infinitely many n.