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By using the Jiang’s function J2 (ω) we prove that there exist infinitely many integers n such that n=2P1, n+1=3P2,, n+k−1=(k+1) Pk are all composites for arbitrarily long k, where P1, P2,…, Pk are all primes. This result has no prior occurrence in the history of number theory.

On the Consecutive Integers n+i-1=(i+1) Pi