September 2016 - mathfun@mailman.xmission.com

[math-fun] Comparing sizes of sets of natural numbers
by James Propp 05 Sep '16

05 Sep '16

I'd welcome math-funsters' comments on http://mathoverflow.net/questions/248994/comparing-sizes-of-sets-of-natural… . Teaser: The size of {0,2,4,6,...} exceeds the size of {1,3,5,7,...} by exactly 1/2. "From a certain point of view." Jim Propp

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[math-fun] Gaa, engine BRAKing!
by Bill Gosper 05 Sep '16

05 Sep '16

Who remembers when we had to break engines in? --rwg

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Re: [math-fun] "Finite" meaning "nonzero"
by Bill Gosper 05 Sep '16

05 Sep '16

Your car's gearbox has the finite gears and neutral. For accelerating, neutral = 0. For engine breaking and roll starting, neutral = ∞. On 2016-09-04 09:11, Tomas Rokicki wrote: > It's an interesting use of the word "finite" in the sense that the > opposite of "finite" probably needs to be "infinite", but here the > opposite appears to be "zero". Further, if we are looking at > a case like this, a "small but non-zero" asymptote implies a > satisfying population for "finite" would have to be, in count, > *infinite*. > > I believe we are in the world of Humpty Dumpty here. > > On Sun, Sep 4, 2016 at 5:54 AM, Veit Elser <ve10(a)cornell.edu> wrote: > >> I raised this point in a Physics Today letter to the editor (1980s) >> and was met with unanimous defense of this usage of “finite”. >> >> -Veit Physicists take logarithms a lot. --rwg >> >> > On Sep 3, 2016, at 9:45 PM, James Propp <jamespropp(a)gmail.com> wrote: >> > >> > Are you using "finite" to mean "nonzero", Bill? >> > >> > I have always disparaged this usage, but maybe I should reconsider my >> > prejudice. What do you all think about this? >> > >> > Jim Propp >> > >> > On Saturday, September 3, 2016, Bill Gosper <billgosper(a)gmail.com >> > <javascript:_e(%7B%7D,'cvml','billgosper(a)gmail.com');>> wrote: >> > >> >> Despite my distaste for decimal digits, I noticed >> >> ... >> >> Is the asymptotic density finite? --rwg

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Re: [math-fun] peculiar sequence
by Keith F. Lynch 04 Sep '16

04 Sep '16

Bill Gosper <billgosper(a)gmail.com> wrote: > Despite my distaste for decimal digits, I noticed (while showing > Mathematica to new user Miles) that 52! ends in twelve 0s both in > decimal and hex. Equality happens ... with gradually declining > frequency: Tomas Rokicki <rokicki(a)gmail.com> wrote: > Maybe it's finite, maybe it's not. Asymptotically of course the > count of zeros in both decimal and hex in n! are n/4; surprising > that. > I could only get to 10^11 quickly: .... Overuse of Mathematica rots the brain. If you want an exact count, the number of 0-digits at the end of n! is sum(floor(n/5^j)) for decimal, and floor(sum(floor(n/2^j))/4) for hexadecimal. In both cases j ranges over all positive integers. Using those formulas, it should be easy to get way past 10^11. "How many zero-digits does one thousand factorial end with" is a fun problem to give to people to solve in their head.

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[math-fun] "Finite" meaning "nonzero"
by James Propp 04 Sep '16

04 Sep '16

Are you using "finite" to mean "nonzero", Bill? I have always disparaged this usage, but maybe I should reconsider my prejudice. What do you all think about this? Jim Propp On Saturday, September 3, 2016, Bill Gosper <billgosper(a)gmail.com <javascript:_e(%7B%7D,'cvml','billgosper(a)gmail.com');>> wrote: > Despite my distaste for decimal digits, I noticed > ... > Is the asymptotic density finite? --rwg > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >

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[math-fun] Depressing behavior of Tribonacci constant
by Neil Sloane 03 Sep '16

03 Sep '16

This constant T = 1.83928... (A058265) is the real root of X^3-X^2-X-1. Its Beatty sequence (or spectrum) is the sequence a(n) = floor(nT), A158919. I found a lovely (conjectured) recurrence for a related sequence, but as a sanity check I applied the same analysis to A158919 itself. What I found is depressing because A158919 is regular for 20000 terms but then a new pattern appears. This sort of thing is common in Pisot sequences (e.g. A007699), but I was suprised to encounter it here. It raises the possibility that my lovely conjecture may be only a mirage. Beatty sequences have been studied since the 1920s, especially in Canada, and there are dozens of paperss. Could someone explain what is going on and point me to the right reference? Start with the Beatty sequence and apply some operations: A158919, floor(nT): 0, 1, 3, 5, 7, 9, 11, 12, ... [I used 100000 terms] DIFF: 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, [DIFF = "differences"] RUNS: 1, 5, 1, 5, 1, 5, 1, 5, 1, 6, 1, 5,... [RUNS = "run lengths"] BISECT: 5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 5, 5, 6, 5,.. [take alternate terms] RUNS: 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3... [now down to 7144 terms] BISECT: 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3,... [3467 terms] RUNS: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... EXCEPT FOR 2's at terms 612, 1224, 1836, 2448, 3060. Help! Presumably the trouble at term 612 comes from the 305 in the continued fraction for T, (A019712): 1, 1, 5, 4, 2, 305, 1, 8, 2, 1,... ?

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[math-fun] peculiar sequence
by Bill Gosper 03 Sep '16

03 Sep '16

Despite my distaste for decimal digits, I noticed (while showing Mathematica to new user Miles) that 52! ends in twelve 0s both in decimal and hex. Equality happens for Select[Range[0, 101], IntegerExponent[#!, 10] == IntegerExponent[#!, 16] &] {0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 20, 21, 22, 23, 28, 29, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 65, 70, 71, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 90, 91, 95, 100, 101} with gradually declining frequency: In[535]:= Count[Range@99999, n_ /; IntegerExponent[#, 10] == IntegerExponent[#, 16] &[n!]] // tim During evaluation of In[535]:= 3610.464197,0 Out[535]= 23617 Is the asymptotic density finite? --rwg

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[math-fun] A091178: consecutive integers
by Zak Seidov 01 Sep '16

01 Sep '16

https://oeis.org/A091178 Are there 100 (even more) consecutive integers? thx, zak

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[math-fun] Palo Alto to enforce zoning ban on software coding
by Henry Baker 01 Sep '16

01 Sep '16

FYI -- [Micro-aggression warning: undecidability lurks below.] Sheesh! Talk about heavy handed; Palo Alto could simply have bounded time and/or space for its Turing Machines. But NOoooh! they've prohibited *stacks* (no CF), and probably even finite state machines. I presume that this is another attack on encryption technology (it IS aimed at Palantir -- which occupies much of Palo Alto -- among other companies), but perhaps that is just paranoia. I believe that the zoning ban also applies to macros -- code-producing code -- and compilers. Perhaps it's time to resurrect Google's Ark -- Google's plan to station a floating high-tech development platform off-shore to get around H1B visa hiring restrictions. http://www.siliconbeat.com/2016/08/31/no-coding-palo-alto-city-takes-silico… No coding in Palo Alto? The birthplace of Hewlett Packard and Xerox Parc and founding place of Facebook is now considering whether to enforce a zoning regulation banning firms whose "primary business is research and development, including software coding," according to the New York Times. Palo Alto Mayor Patrick Burt told the Times: "Big tech companies are choking off the downtown. Its not healthy." "For many of the 66,000 residents of Palo Alto, all this coding and innovation may be a bit too much." "But Palo Alto may not want the jobs either. Making coding off-limits in certain sections, however that is enforced, sends that message." http://www.nytimes.com/2016/08/30/us/spotty-cell-reception-in-the-heart-of-… "the mayor [of Palo Alto] is looking to enforce, in some form, an all-but-forgotten zoning regulation that bans companies whose primary business is research and development, including software coding."

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