It is known that there are infinitely many Sierpinski ′ numbers and Riesel numbers in the sequences of triangular numbers, hexagonal numbers, pentagonal numbers, and many other polygonal sequences. Let Tk denote the kth triangular number. We prove an additional property: for infinitely many k, every integer in the sequence Tk2n + 1 with n a positive integer always has at least two distinct prime divisors. Furthermore, there are infinitely many k such that every integer in the sequence Tk2n 1 with n a positive integer always has at least two distinct prime divisors. Also, there are infinitely many k such that every integer in both sequences Tk2n +1 and Tk2n1 with n a positive integer always has at least two distinct prime divisors. Moreover, the above results hold when replacing Tk with infinitely many di?erent s-gonal number sequences.
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