Quadratic sequences with prime power discriminators
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The discriminator of an integer sequence s = (s(i))_i ≥ 0, introduced by Arnold, Benkoski, and McCabe in 1985, is the function D_s (n) that sends n to the least integer m such that the numbers s(0), s(1), …, s(n - 1) are pairwise incongruent modulo m. In this note, we try to determine all quadratic sequences whose discriminator is given by p^⌈log_p n ⌉ for prime p, i.e., the smallest power of p which is ≥ n. We determine all such sequences for p = 2, show that there are none for p ≥ 5, and provide some partial results for p = 3.
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