More experiments: I tried using g[n_] := Exp[Sqrt[n]] with the same form of K: K[max_] := For[n=1,n<=max,n=n+1,If[N[Abs[FractionalPart[g[n]]-1/2],3] > .49, Print[n," -> ",AccountingForm[N[g[n],70]]],Null]] ... and even plugging in 1 million got no output at all. Even though random numbers should be hits 2% of the time! Then, tried instead this definition: g[n_] := Exp[EulerGamma*Sqrt[n]] with the same K: K[max_] := For[n=1,n<=max,n=n+1,If[N[Abs[FractionalPart[g[n]]-1/2],3] > .49, Print[n," -> ",AccountingForm[N[g[n],70]]],Null]], and this was the output up to 1000: In[89]:= K[1000] 44 -> 46.00801254989580374866821393281718990610808755385916137088675721910220 72 -> 133.9996690682195024916379108364436984642366431675814385787629532249629 106 -> 380.9998560899922652985048364133353260948427549094148511587819994344508 142 -> 970.9922482270692677823683566362486703758343384197035855953939483686496 152 -> 1231.998165320086794113206659448727243056884763108326053348929991822208 191 -> 2913.991881693797749913437687584167601069437667150760604838637626844605 318 -> 29532.00302716531235593173259733035039994363373691206154022508694551477 320 -> 30502.00551977682189313224051223001203289587620722158978081666039302332 345 -> 45311.00106373947342773854350867556874544213023995930362766218155941872 454 -> 219452.9945686432069421577154825428216777397159357514383664094397968059 466 -> 257911.9921461867934643982746851372951258497151014207816190016716897832 503 -> 418987.9922146247967498624482658586339260085392808280647138753327451585 525 -> 554416.9963737390879335581569231667971310399032797345458721163129398121 674 -> 3221604.998395203239131677759219596409276001991060563008032795557771631 688 -> 3761111.003683565245331237337931653954132556761830043597215507655483516 741 -> 6666259.991745126708920504918284533297983410319524424920971083230722710 757 -> 7891525.001284764658545293638044726097008102685225884136766957587145273 765 -> 8580446.004732329867080747545028212880152161498187621710548761089218119 793 -> 11461918.99161594923239240165170113390157236484193567263737374380053738 803 -> 12694854.00796358714126243803067300233356698650720543389962885867148105 870 -> 24776745.99653706667749829322322634885572464342147049948027495645265893 927 -> 42896361.99301774554623061821047007944328721040711227839052158844485217 ------------------------------------------------------------------------------ Which is 22 hits, pretty close to 2% of 1000 -- though there are still more just-belows (14) than just-aboves (8). (Going through 3000 instead of 1000, there are a total of 24 just-aboves and 38 just-belows, so this may beg for an explanation. There are 25 total hits 1001-2000, and 15 total hits 2001-3000.) Okay, so why does exp(sqrt(n)) have so few almost-integers -- a total of 0 up through 1 million if you define them to be within .01 of an integer, whereas 2% of 1 million is 20000 ????? (Assuming my Mma didn't have a bug.) --Dan