Dan A> What's enigmatic about this kind of very simple expression for a near-integer is whether it is a coincidence, or whether it begs for an explanation. To explain or not to explain: that is the question. For example, it's fairly well known that exp(pi*sqrt(163)) is an integer up to < 10^(-12), and this can be explained by the q-expansion of the j-invariant. This explanation also holds for the near-integers exp(pi*sqrt(67), up to < 10^(-5), exp(pi*sqrt(43), up to < 10^(-3), exp(pi*sqrt(19), up to < 1/4. Wikipedia also mentions this amazing discovery of D. Bailey & J. Borwein: Integral_{x=0 to oo} (cos(2x) * Prod_{n=1 to oo} cos(x/n)) dx = pi/8, up to < 10^(-42). Wikipedia claims this is just a coincidence, but I'd call that chutzpah. --Dan Me too. To numerically investigate this, the infinite product requires drastic acceleration. A completely inadequate but mildly interesting acceleration follows. Starting with the middle form in Out[153]= {Sqrt[Product[Sinc[x/(k-1/2)], {k, n}]^3/Product[Cos[x/k], {k, n}]], Product[Cos[x/k], {k, 2*n}], Product[Sinc[x/(k-1/2)], {k, 2*n}]} Convert to the third form by the usual differentiate - Mittag-Leffler - sumswap, giving a negligible speedup. Combine weighted powers to get the first form, and take half as many terms to reflect the doubled complexity. It still wins: In[157]:= Table[Evaluate[%153 /. x -> .69], {n, 3^Range[5]}] Out[157]= {{0.66406, 0.685026, 0.669198}, {0.660881, 0.668976, 0.663342}, {0.660477, 0.663315, 0.661396}, {0.66043, 0.661393, 0.660748}, {0.660425, 0.660748, 0.660532}} --rwg