April 2005 - mathfun@mailman.xmission.com

[math-fun] Puzzle about polynomial roots
by Daniel Asimov 14 Apr '05

14 Apr '05

For n >= 1 let E_n(z) = 1 + z + z^2/2! + ... + z^n/n!, the truncated series for exp(z). Let {r(n,1),...,r(n,n)} be the set of roots of E_n(z). (It's easy to verify there are no repeated roots.) Puzzle: Find the limit as n -> oo of sum from k = 1 to n of 1/(E_n)'(r(n,k)). --Dan

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[math-fun] sum 1/p
by Richard Schroeppel 14 Apr '05

14 Apr '05

Neil Sloane posted this on the NMBRTHRY list. I'm replying here because this seems to be a programming problem as well as a math and number theory problem. Date: Fri, 25 Mar 2005 08:56:33 -0500 From: "N. J. A. Sloane" <njas(a)research.att.com> Subject: Sum of reciprocals of primes To: NMBRTHRY(a)LISTSERV.NODAK.EDU I keep seeing quotes that the sum of 1/p for all the primes p that mankind has ever seen will never exceed 4. Just how hard would it be to compute the next term in these two sequences? Neil Sloane %I A016088 %S A016088 5,277,5195977 %N A016088 a(n) = smallest prime p such that Sum_{ primes q = 2, ..., p} 1/q exceeds n. %C A016088 The indices of these primes are in A046024: a(n) = A000040(A046024(n)). %D A016088 Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Plenum Press, NY and London, 1996, page 64. %H A016088 <a href="http://www.research.att.com/~njas/sequences/Sindx_1.html#1overn">Index entries for sequences related to decimal expansion of 1/n</a> %t A016088 s =0; k =1; Do[ While[ s =N[ s +1/Prime[ k ], 36 ]; s <= n, k++ ]; Print[ Prime[ k ] ]; k++,{n,1,3} ] %Y A016088 Cf. A046024. %K A016088 nonn,bref,nice,hard,more %O A016088 1,1 %A A016088 Robert G. Wilson v (rgwv(AT)rgwv.com) %E A016088 a(3) corrected by Ulrich Schimke (UlrSchimke(AT)aol.com) who estimates that a(4) is approximately 1.8*10^18. %I A046024 %S A046024 3,59,361139 %N A046024 a(n) = smallest k such that Sum_{ i = 1..k } 1/prime(i) exceeds n. %C A046024 The corresponding primes prime(a(n)) are in A016088. %C A046024 Index m for which the prime harmonic number p[ m ] := Sum[ 1/Prime[ k ],{k,1,m} ] >= n. %H A046024 E. W. Weisstein, <a href="http://mathworld.wolfram.com/PrimeNumber.html">Link to a section of The World of Mathematics.</a> %H A046024 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicSeriesofPrimes.html">Harmonic Series of Primes</a> %Y A046024 Cf. A004080, A16088. %K A046024 nonn,more,bref,nice %O A046024 1,1 %A A046024 Eric W. Weisstein (eric(AT)weisstein.com) It would be straightforward to set up a distributed computation to sum 1/p for p < 1.8e18, and locate the specific P that bumps the sum over 4. In a few more years the resources will be available to many of us as individuals. I've thought about doing similar computations myself, but always get bogged down in "As long as you're touching all those primes, what else do you want to compute?". I also want sum 1/(p-1), product P, etc. For the specific problem of finding the 4-crossing, it may not be necessary to touch every prime: group the primes into small blocks, whose reciprocals are nearly equal, and compute the upper & lower bounds on the reciprocal sum. Does this save any work? Finding A(5) will be more challenging: the terms grow roughly as Eth powers, so A(5) will be somewhere around (1.8 * 10^18) ^ 2.718 ~= 10^50. Absent some theory advances, we won't be computing that sum anytime soon. Odlyzko & friends have pushed exact computations of pi(N) up to around N = 10^21 using a zeta function based formula, so perhaps sum(1/p) could be tackled similarly. This doesn't require touching every prime. Sequence comment: The Sequence database contains many related sequences such as A016088 and A046024. They are usually cross referenced, but it would be nice to do a little better. In this case, if I started with A046024, I might conclude that I didn't need to look at A016088, since the contents were obvious and I wasn't interested in the exact numbers. This would cause me to miss the little note at the end of A016088 about the estimate for A(4). Both sequences also omit the asymptotic formula for sum 1/p, which is something like gamma + loglog p. Wish: When several related sequences are in the database, designate one of them as Main, and include a short article about all of them and the relationships. Then each related sequence can point to the Main one. This would be especially helpful for sequences which count groups of different kinds. Rich rcs(a)cs.arizona.edu

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[math-fun] Unsolved? problem
by Richard Guy 13 Apr '05

13 Apr '05

I'm collecting Murray Klamkin problems and solutions and am currently going thru Math Mag. I came across Problem 886, Math Mag 48(1975) 57--58 [nothing to do with Murray] which isn't properly stated but should read as in OEIS A003508 : a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1). This leads to 1,2,3,4,7,8,11,12,18,24,30,41,42,55,... The original problem asked that if you start elsewhere, e.g., 5,6,12, ... or 9,13,14,24, ... or 10,18, ... or 15,24, ... do you always merge with the original sequence? Evidently 91,112,122,186,... takes a little while. Has anyone ... Can anyone prove Charles Trigg's guess ? R.

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[math-fun] Unsolved? problem
by Chris Landauer 13 Apr '05

13 Apr '05

hihi, all - rkg(a)cpsc.ucalgary.ca wrote ... > [define a sequence by] > a(n) = a(n-1) + 1 + sum of > distinct prime factors of > a(n-1) that are < a(n-1). > This leads to > 1,2,3,4,7,8,11,12,18,24,30,41,42,55,... > The original problem asked that > if you start elsewhere, e.g., > 5,6,12, ... or > 9,13,14,24, ... or > 10,18, ... or > 15,24, ... > do you always merge with the > original sequence? Evidently > 91,112,122,186,... takes a little while. ... i wrote a simple program to compute these sequences, but only with 1003 primes, and all of the sequences merged with the original a[0]=1 sequence up to a[0]=393 (starting with a[0]=91 takes 24 terms, starting with a[0]=105 takes 41, both local maxima) presumably this can be easily extended more later, cal

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Re: [math-fun] sum 1/p
by N. J. A. Sloane 13 Apr '05

13 Apr '05

Rich said: Wish: When several related sequences are in the database, designate one of them as Main, and include a short article about all of them and the relationships. Then each related sequence can point to the Main one. Me: I totally agree, and would welcome such articles! As a watered-down version of this idea, I often build an entry in the Index to the OEIS (there's a link at the top page) that collects all related sequences. Example: Barnes-Wall lattices, <a NAME="BW">sequences related to (start):</a> Barnes-Wall lattices, groups of: A014115*, A014116* Barnes-Wall lattices, in 16-dim., theta series of: A008409* Barnes-Wall lattices, kissing numbers of: A006088* Barnes-Wall lattices, odd: A014711* Barnes-Wall lattices, see also A035596 Barnes-Wall lattices: see also <a href="http://www.research.att.com/~njas/sequences/Sindx_Cl.html#cliff">Clifford groups</a> and then each sequence in the group has a pointer to that Index entry, like this: %H A014115 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ba.html#BW">Index entries for sequences related to Barnes-Wall lattices</a> I would welcome further Index entries like that too. Neil Sloane

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[math-fun] concatenated primes
by David Wilson 13 Apr '05

13 Apr '05

This problem came out of a problem being worked on by Gary Darby. Is there an infinite sequence of positive integers whose distinct pairwise concatenations are all primes? - David W. Wilson "Truth is just truth -- You can't have opinions about the truth." - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"

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[math-fun] Martin Gardner Collection
by Bernie Cosell 12 Apr '05

12 Apr '05

I noticed in the MAA Spring book catalog that they're offering "The Entire Collection of his Scientific American Columns on two CDs" /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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RE: [math-fun] sequences from Hilbert's 16th problem?
by Daniel Asimov 09 Apr '05

09 Apr '05

Hi, Neil. I shouldn't be encouraging you to put "erroneous" sequences in OEIS, but it's your baby, so . . . here is Ilyashenko's e-mail address: yulijs(a)mccme.ru Regards, Dan

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[math-fun] New methods in combinatorial game theory
by David Gale 06 Apr '05

06 Apr '05

For people interested in such things, the paper linked below solves some previously unsolved problems for the game of Chomp. For example, it proves that the 3 by n game always has a unique first move. The interest, however, is not so much in the specific results as in the innovative methodology which uses the "renormalization" technique (a la Feigenbaum), -totally different from anything I've seen applied to combinatorial games. For Chomp-savvy people, the paper gives precise estimates for the limit size of the first moves ("bites") and shows that the relative frequency of one-row to two-row initial bites is sqrt(2)-1 to 2-sqrt(2). There are intriguing illustrations. ftp://ftp.orie.cornell.edu/pub/techreps/TR1422.pdf David

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RE: [math-fun] "Random" Deck? --> OEIS suggestion
by N. J. A. Sloane 02 Apr '05

02 Apr '05

Dan: Derangements are A000166. Dividing by factorials gives 1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, 16687/45360, 16481/44800, 1468457/3991680, ... which is A053557/A053556, with references going back to Euler. Neil -------------------- Dan said: > Still: Neil -- if you're out there -- the disordered hats problem > suggests two new sequences which a search on the word > "derangement" did not find in the OEIS: > > a) the numerators, and > > b) the denominators > > of the (always rational) probability (in lowest terms) that a > random permutation of n items is a derangement.

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