THE PRIMES IN SMARANDACHE POWER PRODUCT SEQUENCES

For any positive integer k, let Ak be the Smarandache k -power product sequence. In this paper we prove that if k is an odd integer, \\ith k> 1, then Ak contains only one prime 2. In [1], Iacobescu defined the sequence {H-ciC2 ... Cn}n~l; is the Smarandache cubic product sequence, where Cn is the n-th cubic number. Simultaneous, he posed the following question: Question: Hou many primes are in the sequence {I +CIC2 ... CJn~1 ? We nou give a general definition as follows: For any positive integers k, n let and let Ak = {ak(n)ln~l. Then Ak is called the Smarandache k-power product sequence. In this paper we prove the following result: Theorem. If k is an odd integer, with k> 1, then Ak contains only one prime 2. Clearly, the above result completely solves Iacobescu's question. Proof of Theorem. We see from (1) that Ifk is an odd integer, with k>l, then from (2) we get (3) akCn)=lk-Cn!)k = (l+n!)(1 n! 7 (nl)2 _ ... _ (nf)k-Z", (n,)k-l) When n = I, ak(l) = 2 is a prime. When n > 1, since we find from (3) that ak(n) is not a prime. Thus, the sequence Ak contains only one prime 2. The theorem is proved. Reference: 1. F. Iacobescu, "Smarandache partition type and other sequences", Bulletin of Pure and applied Sciences, 16E( 1997), No.2, 237-240.