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EXAMPLE
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Type 2 prime p = 5: the sequence of 5-adic valuations [v_5(a(n)) : n = 1..100] = [0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 14, 12, 13, 12, 12, 14, 13, 14, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 19, 18, 19, 18, 18, 21, 19, 20, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25].
Note that v_5(a(100)) = 25 = 100/(5 - 1), in agreement with the asymptotic behavior for type 2 primes conjectured above.
Type 3 prime p = 7: the sequence of 7-adic valuations [v_7(a(n)) : n = 1..100] = [0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0], showing the oscillatory behavior for type 3 primes conjectured above.
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