September 2014 - mathfun@mailman.xmission.com

Re: [math-fun] highly composite numbers
by Warren D Smith 02 Sep '14

02 Sep '14

OK, I did a computation. Letting d(N) denote the number of divisors of N, I find 1. for 0<N<e^380 that d(N)<2.44*2^Li(lnN) with tightness only for d(6)=4. Does this upper bound work forever? No. In an excellent example of misleading behavior for small N, I found d(N)>1591396267*2^Li(lnN) for a certain N with lnN =approx= 1000010.9002377. 2. All highly composite numbers N with 1<N<e^6969 obey d(N)>1.12*2^Li(lnN), with tightness only for d(45360)=100. This supports my conjecture that d(N)>=2^Li(lnN) for all N>0 (which is tight when N=1).

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[math-fun] wholesale reciprocals
by rcs＠xmission.com 02 Sep '14

02 Sep '14

Question for the old-timers: I'm trying to pin down a (possibly faux) memory from the 1960s. Back in the day, floating point division was a luxury-- frequently missing from the instruction set, handed off to a subroutine, in any case very slow. If your program used it, it was practically guaranteed to be the slow step. One way to reduce the number of FDVs was to calculate several reciprocals at a time, and then use multiplications to calculate the needed quotients. The trick for calculating two reciprocals at once was to notice that 1/(XY) takes one FDV, and then you can get 1/X = X * 1/(XY) and 1/Y similarly. The trick easily generalizes to several reciprocals, changing N reciprocals into 1, plus 3(N-1) multiplies. (If you want N quotients, you pay another N multiplies.) There's also a minor cost in numerical accuracy, using several calculations (with their attendant rounding error), where one would do. Also, for the careful, the analysis of whether intermediate results like XYZW might over/underflow. It's my impression that I'd seen this trick mentioned in a couple of places, possibly Organick's Fortran book, or Numerical Algorithms in Fortran, and also in actual code. Can anyone confirm or refute my fuzzy recollections? Rich

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Re: [math-fun] dragon gif
by Bill Gosper 02 Sep '14

02 Sep '14

Jörg>* Bill Gosper <billgosper(a)gmail.com> [Sep 01. 2014 08:00]: [...] Holy cr@p, http://gosper.org/semizerp.htm andhttp://gosper.org/semizerp1.htm on steroids! Another 13fold recursion, replete with inhomogeneous scaling: Mandelbrot's recursive Snowflake. Can such be likewise "teardropped"? He also had a 7fold recursion, I think. --rwg Note I computed _all_ curves generated by L-systems with just one non-constant symbol.< Staggering. So somewhere in all that are these three "sevens" gosper.org/IMG_0245.JPG gosper.org/IMG_0246.JPG gosper.org/IMG_0247.JPG (hardcopied in the 70s, pre-laser.) Also, did you notice this way of mis-teardropping the France fractal? gosper.org/jelly7.bmp Lastly, I had forgotten about these two teardroppings of Mandelbrot's seven Snowflake recursion: gosper.org/trozeImage8.gif gosper.org/trozeImage13.gif Maybe someone can do his thirteen, but I can't do it in my head. I just found this: http://www.meden.demon.co.uk/Fractals/frozenteardrop.html . Jörg> As different curves can give the same overall shape, let's count the shapes (by grid and order): The example pictures are by filename (given in http://jjj.de/tmp-extra/ ). Triangular grid, only turns of 120 degs: c3/search-r03-curves.txt: 1 c3/search-r04-curves.txt: 1 c3/search-r07-curves.txt: 3 c3/search-r09-curves.txt: 5 c3/search-r12-curves.txt: 10 c3/search-r13-curves.txt: 15 c3/search-r16-curves.txt: 17 c3/search-r19-curves.txt: 71 c3/search-r21-curves.txt: 212 c3/search-r25-curves.txt: 184 c3/search-r27-curves.txt: 543 c3/search-r28-curves.txt: 842 c3/search-r31-curves.txt: 1848 Example (order 19): all-r19-curves.pdf all-r19-tiles.pdf Example (order 16): all-r16-curves.pdf all-r16-tiles.pdf Cf. https://oeis.org/A234434 Rectangular grid (turns by 90 degs): c4/search-r05-q-curves.txt: 1 c4/search-r09-q-curves.txt: 1 c4/search-r13-q-curves.txt: 4 c4/search-r17-q-curves.txt: 6 c4/search-r25-q-curves.txt: 33 c4/search-r29-q-curves.txt: 39 c4/search-r37-q-curves.txt: 164 c4/search-r41-q-curves.txt: 335 c4/search-r49-q-curves.txt: 603 Example (order 25): all-r25-q-curves-sty1.pdf all-r25-q-tiles-sty1.pdf Any number of curves on the same grid can be "multiplied" (noncommutatively). Figures like http://gosper.org/semizerp.htm can be obtained from both families above by a sort of "Kronecker division", for arbitrary divisor. Triangular grid, containing turns of 60 degs: c6/search-r07-b-curves.txt: 1 c6/search-r13-b-curves.txt: 3 c6/search-r19-b-curves.txt: 7 c6/search-r25-b-curves.txt: 10 c6/search-r31-b-curves.txt: 63 c6/search-r37-b-curves.txt: 157 c6/search-r43-b-curves.txt: 456 c6/search-r49-b-curves.txt: 1830 Example (order 49): all-r49-b-tile-plus-shapes-2.pdf all-r49-b-tile-plus-shape-borders-2.pdf See the files side by side. There are far more curves (with higher orders more and more tend to give the same shape). The above corresponds to roughly a quarter million curves. A rather random collection from the beginning phase of my search (including curves with more than one non-constant symbol) is http://jjj.de/3frac/ Very many of them where created with pencil and paper. Best, jj P.S.: http://dimacs.rutgers.edu/Workshops/OEIS/abstracts.html#arndt < Break a leg. --rwg

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[math-fun] highly composite numbers
by Warren D Smith 02 Sep '14

02 Sep '14

If N has more divisors than any lesser number, N is "highly composite." http://oeis.org/A002182 I conjecture that the number d(N) of divisors of any highly composite N always obeys d(N) >= 2^Li(ln(N)). Results of S.Ramanuan: Highly composite numbers, Proc. London Math. Soc. (2) 14 (1915) 347-409. http://ramanujan.sirinudi.org/Volumes/published/ram15.pdf show that my conjecture is true at least for an infinite subset of the highly composite N. It looks plausible that by similar techniques you can prove it for all sufficiently large highly composite N, although as far as I know nobody has done so. If so, and if we knew exactly what "sufficiently large" meant, that'd reduce the conjecture to a finite computation. Which then might be within reach of computer (e.g. at least the first 120000 highly composite numbers are known). -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] dragon gif
by Bill Gosper 01 Sep '14

01 Sep '14

Jörg>Telepathy? If have been thinking about _exactly_ this for some time, but did not code it up because my home-grown software does not really make it easy... Extra points if you do that gif-ery for the curve http://jjj.de/tmp-ryde/t-shirt-R13-15-hires-cropped.png (L-system at bottom, turns are by 120 degrees). Total curve should be light gray, and the moving part should be 1/13 of the total (so the same shape will appear 13 times, as indicated in the image). ... plus you get a T-shirt! (If you use the image to, well, print one. I got one and it looks quite OK). Best, jj Holy cr@p, http://gosper.org/semizerp.htm and http://gosper.org/semizerp1.htm on steroids! Another 13fold recursion, replete with inhomogeneous scaling: Mandelbrot's recursive Snowflake. Can such be likewise "teardropped"? He also had a 7fold recursion, I think. --rwg * Bill Gosper <billgosper(a)gmail.com> [Aug 31. 2014 11:27]: The Dragon *function* (not curve, dammit) continuously mapping [0,1] into the box[-1/3-i/3, 7/6+2i/3] with the image of a circulating interval of length 1/10 highlighted, pausing at various multiples of 1/10 to show anatomical measurements. gosper.org/dragondrift.gif --rwg

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