September 2008 - mathfun@mailman.xmission.com

[math-fun] Asymptotics of Moebius function
by N. J. A. Sloane 10 Sep '08

10 Sep '08

Dear Math Fun, Does anyone know the asymptotics for this sequence? (I did a crude estimate, but I have no confidence in it.) Thanks! Neil %I A070548 %S A070548 1,1,1,1,1,2,2,2,2,3,3,3,3,4,5,5,5,5,5,5,6,7,7,7,7,8,8,8,8,8,8,8,9,10, %T A070548 11,11,11,12,13,13,13,13,13,13,13,14,14,14,14,14,15,15,15,15,16,16,17, %U A070548 18,18,18,18,19,19,19,20,20,20,20,21,21,21,21,21,22,22,22,23,23,23,23 %N A070548 a(n) = Card{ k in range 1<=k<=n such that Moebius(k)=1 }. %C A070548 Moebius(k)=1 iff k is the product of an even number of distinct primes (cf. A008683). See A057627 for Moebius(k)=0. %H A070548 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b070548.txt">Table of n, a(n) for n = 1..10000</a> %o A070548 (PARI) for(n=1,150,print1(sum(i=1,n,if(moebius(i)-1,0,1)),",")) %Y A070548 Cf. A008683. Equals A072613(n) + 1. %K A070548 easy,nonn %O A070548 1,6 %A A070548 Benoit Cloitre (benoit7848c(AT)orange.fr), May 02 2002

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[math-fun] ceiling function
by James Propp 10 Sep '08

10 Sep '08

Using the well-orderedness property of the non-negative integers, plus the usual axioms of R as an ordered field and Z as an ordered ring, one can show that the the Archimedean property ("For all real x there exists an integer n with n > x") implies the existence of the ceiling function ("For all real x there exists a smallest integer n with n greater than or equal to x"). But does one need well-orderedness (or equivalently the axiom of induction) to prove this? That is, from the assumption that the ceiling function exists, can one derive the axiom of induction? It seems to me that the proposition that the ceiling function exists has an intermediate character: it cannot be derived from the ordered field and ordered ring axioms, but neither is it strong enough to imply induction. Are there a couple of nice non-standard models of (R,Z,0,1,+,x,>) that make this completely patent? Jim

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[math-fun] New Mersenne Primes
by Guy Haworth 09 Sep '08

09 Sep '08

The GIMPS at www.mersenne.org announces that the M(p) offered up as 'prime' on 2008-08-23 has been verified prime by the first of two confirmations. Good chances that this will be the first prime of over 10m decimal digits: 2nd confirmation due tomorrow, 2008-09-10. Prize time! The M(p) offered up as prime on 2008-09-06, just 14 days later, is scheduled to be confirmed or denied on 2008-09-11: I hope it's bigger or there will be confusion. I guess there's a 'no overtaking' rule on the Mersenne Motorway. Guy p.s. M(M(31)) remains the largest Double Mersenne Number (obviously with prime index, in this case M(31)) whose primality status is known - it is composite. It has been on the top of the DMN pile for 25 years plus!

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[math-fun] 45th and 46th Known Mersenne Primes Found!!!!???
by Robert Baillie 09 Sep '08

09 Sep '08

45th and 46th Known Mersenne Primes Found!!!!??? On August 23rd, a computer reported finding a new Mersenne prime to the server! Because I was on vacation, verification did not begin until the 26th. Two verification runs are in progress. Estimated completion is September 10th or 11th. Check back here for updates! Amazingly, on September 6th, another computer claims finding a new Mersenne prime!! Independent verification has begun. look for updates at: http://www.mersenne.org/

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[math-fun] Pretty iterated complex log picture
by Steve Witham 07 Sep '08

07 Sep '08

Hi-- I wanted to plot iterations of the complex log, and used it as an excuse to start to learn the ReportLab pdfgen Python library. I really like this first result: http://www.tiac.net/~sw/2008/09/Iterate_Log/ I know you Mathematica, Maple, Maxima, etc. people can just *gesture* at your computers and they will do this in 3D and color, but I like to turn the gears with my hands. --Steve "Movin' up to the double-M, two thousand..." --Soul Coughing

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Re: [math-fun] Distributivity of addition over multiplication
by Dan Asimov 07 Sep '08

07 Sep '08

<< You asked if there was an intuitive explanation as to why the satisficing set is 1 = x + y + z. >> Actually that is only part of the solution set. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Distributivity of addition over multiplication
by Dan Asimov 07 Sep '08

07 Sep '08

Jason wrote: << On Thu, 21 Aug 2008, Dan Asimov wrote: > Of course > > x + yz = (x + y)(x + z) > > is not true in general for (x,y,z) in R^3. > [...] Interesting! I plotted each side and derived the intersection plane, but it does indeed seem quite odd that that would be the resulting shape. What's the answer? >> That was 17 days ago. So as Gertrude Stein is reputed to have said, "What is the question?" --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Distributivity of addition over multiplication
by Dan Asimov 07 Sep '08

07 Sep '08

Of course x + yz = (x + y)(x + z) is not true in general for (x,y,z) in R^3. But I don't think I would have guessed which (x,y,z) in R^3 that it does hold for. It only takes a few seconds to figure out what this set is. But can someone give an intuitive reason for why it is what it is? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Concatenate sum of neighbouring digits + iterate
by Eric Angelini 05 Sep '08

05 Sep '08

Hello Math-Fun, this is perhaps old hat, I don't know... Take an integer "a" like 962 Sum the first pair of digits: 9+6=14 Sum the next pair: 6+2=8 Concatenate the results: 148 Iterate: 148 --> 512 --> 63 --> 9 Take "b" = 999 999 --> 1818 --> 999 Are there many "looping" integers like 999? Best, É.

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[math-fun] Fwd: possible new mersenne prime
by rcs＠xmission.com 05 Sep '08

05 Sep '08

I recall a proposal to save a near-the-end Lucas Test value, which would allow a quick probable-up-or-down check by repeating the last several steps. The notion is that it's hard to fake this by working backwards a few steps. Rich ----- Forwarded message from rjbaillie(a)frii.com ----- Date: Fri, 05 Sep 2008 07:22:32 -0400 From: Robert Baillie <rjbaillie(a)frii.com> Reply-To: Robert Baillie <rjbaillie(a)frii.com> Subject: possible new mersenne prime To: Richard Schroeppel <rcs(a)CS.Arizona.EDU> hi rich, i just happened to look at the site yesterday: www.mersenne.org bob --- 45th Known Mersenne Prime Found??? On August 23rd, a computer reported finding a new Mersenne prime to the server! Because I was on vacation, verification did not begin until the 26th. Two verification runs are in progress. Estimated completion is September 12th and September 16th. Check back here for updates! ----- End forwarded message -----

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