科研之友
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摘要
Let n, k, a and c be positive integers and b be a nonnegative integer. Let nu(2)(k) and s(2)(k) be the 2-adic valuation of k and the sum of binary digits of k, respectively. Let S(n,k) be the Stirling number of the second kind. It is shown that nu(2)(S(c2(n), b2(n+1) + a)) >= s(2)(a) - 1, where 0 < a < 2(n+1) and 2 t c. Furthermore, one gets that nu(2)(S(c2(n), (c - 1)2(n) + a)) = s(2)(a) - 1, where n >= 2, 1 <= a <= 2(n) and 2 c. Finally, it is proved that if 3 <= k <= 2(n) and k is not a power of 2 minus 1, then nu(2)(S(a2(n), k) - S(b2(n),k)) = n + nu(2)(a - b) inverted right perpendicularlog(2), kinverted left perpendicular + s(2)(k) + delta(k), where delta(4) = 2, delta(k) = 1 if k >4 is a power of 2, and delta(k) = 0 otherwise. This confirms a conjecture of Lengyel raised in 2009 except when k is a power of 2 minus 1.
