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Denoting by pn and cn the nth prime number and the nth composite number, respectively, we prove that both the sequence (xn)n≥1, defined by xn=∑k=1n (ck+1−ck) / k−pn / n, and the series ∑n=1∞ (pcn−cpn) / npn are convergent

Sequences and series involving the sequence of composite numbers