Search: id:A000792
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%I A000792 M0568 N0205
%S A000792 1,1,2,3,4,6,9,12,18,27,36,54,81,108,162,243,324,486,729,972,1458,2187,
%T A000792 2916,4374,6561,8748,13122,19683,26244,39366,59049,78732,118098,177147,
%U A000792 236196,354294,531441,708588,1062882,1594323,2125764,3188646,4782969
%N A000792 a(n) = max{ (n-i)a(i) : i<n}; a(0) = 1.
%C A000792 Numbers of the form 3^n, 2*3^n, 4*3^n with a(0)=1 prepended.
%C A000792 If a set of positive numbers has sum n, this is the largest value of 
               their product.
%C A000792 In other words, maximum of products of partitions of n: maximal value 
               of Product k_i for any way of writing n = Sum k_i. To find the answer, 
               take as many of the k_i's as possible to be 3 and then use one or 
               teo 2's (see formula lines below).
%C A000792 a(n) is also the maximal size of an Abelian subgroup of the symmetric 
               group S_n. For example when n =6, one of the Abelian subgroups with 
               maximal size is the subgroup generated by (123) and (456), which 
               has order 9. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
%C A000792 Also the maximum number of maximal cliques possible in a graph with n 
               vertices (cf. Capobianco and Molluzzo). - Felix Goldberg (felixg(AT)tx.technion.ac.il), 
               Jul 15 2001. [Corrected by Jim Nastos and Tanya Khovanova, Mar 11 
               2009]
%C A000792 Every triple of alternate terms {3*k, 3*k+2, 3*k+4} in the sequence forms 
               a G.P. with first term 3^k and common ratio 2. - Lekraj Beedassy 
               (blekraj(AT)yahoo.com), Mar 28 2002
%C A000792 For n>4, a(n) is the least multiple m of 3 not divisible by 8, for which 
               omega(m)<=2 and sopfr(m)=n. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Apr 24 2003
%C A000792 Maximal number of divisors that are possible amongst numbers m such that 
               A080256(m)=n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Oct 13 2003
%C A000792 Or, numbers of form 2^p*3^q with p <= 2, q>=0 and 2p+3q=n. Largest number 
               obtained using only the operations +,* and () on the parts 1 and 
               2 of any partition of n into these two summands where the former 
               exceeds the latter. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 
               07 2005
%C A000792 a(n) is largest number of complexity n in sense of A005520. - David W. 
               Wilson (davidwwilson(AT)comcast.net), Oct 03 2005
%C A000792 a(n) corresponds also to the ultimate occurrence of n in A001414 and 
               thus stands for the highest number m such that sopfr(m)=n, for n>
               =2. - Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 29 2002
%C A000792 A007600(A000792(n)) = n; Andrew Chi-Chih Yao attributes this observation 
               to D. E. Muller. - Vince Vatter (vince(AT)mcs.st-and.ac.uk), Apr 
               24 2006
%D A000792 B. R. Barwell, Cutting String and Arranging Counters, J. Rec. Math., 
               4 (1971), 164-168.
%D A000792 B. R. Barwell, Journal of Recreational Mathematics, "Maximum Product": 
               Solution to Prob. 2004;25(4) 1993 Baywood NY.
%D A000792 R. Bercov and L. Moser, On Abelian permutation groups, Canad. Math. Bull., 
               8 (1965), 627-630.
%D A000792 M. Capobianco and J. C. Molluzzo, Examples and Counterexamples in Graph 
               Theory, p. 207. North-Holland: 1978.
%D A000792 Tomislav Doslic, Maximum Product Over Partitions Into Distinct Parts, 
               Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.8.
%D A000792 S. L. Greitzer, International Mathematical Olympiads 1959-1977, Prob. 
               1976/4 pp. 18;182-3 NML vol. 27 MAA 1978
%D A000792 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 
               2004; p. 396.
%D A000792 P. R. Halmos, Problems for Mathematicians Young and Old, Math. Assoc. 
               Amer., 1991, pp. 30-31 and 188.
%D A000792 E. F. Krause, "Maximizing The Product of Summands", Mathematics Magazine, 
               MAA Oct 1996, Vol. 69, no. 5 pp. 270-271.
%D A000792 L. C. Larson, Problem-Solving Through Problems. Problem 1.1.4 pp. 7. 
               Sprnger-Verlag 1983.
%D A000792 J. W. Moon and L. Moser, On cliques in graphs, Israel J. Math. 3 (1965), 
               23-28.
%D A000792 D. J. Newman, A Problem Seminar. Problem 15 pp. 5;15. Spinger-Verlag 
               1982.
%D A000792 D. A. Rawsthorne, How many 1's are needed?, Fib. Quart. 27 (1989), 14-17.
%D A000792 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000792 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000792 A. C.-C. Yao, On a problem of Katona on minimal separating systems, Discrete 
               Math., 15 (1976), 193-199.
%H A000792 T. D. Noe, <a href="b000792.txt">Table of n, a(n) for n=0..500</a>
%H A000792 MathPro, 20000 Problems Under the Sea, <a href="http://nt3.ftm.ks.edu.tw/
               alzing/10000/14851-14900.htm">Problem 14856.Putnam 1979/A1</a>
%H A000792 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000792 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000792 J. Scholes, <a href="http://www.kalva.demon.co.uk/putnam/psoln/psol791.html">
               40th Putnam 1979 Problem A1</a>
%H A000792 J. Scholes, <a href="http://www.kalva.demon.co.uk/imo/isoln/isoln764.html">
               18th IMO 1976 Problem 4</a>
%H A000792 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A000792 a(n) = 3*a(n-3) if n>4. G.f.: (1+x+2x^2+x^4)/(1-3x^3). - Henry Bottomley 
               (se16(AT)btinternet.com), Nov 29 2001
%F A000792 a(3n) = 3^n; a(3n+1) = 4*3^(n-1) for n>0; a(3n+2) = 2*3^n.
%F A000792 a(n) = if n<=2 then n else a(n-1)+Max{GCD(a(i), a(j))| 0<i<j<n}. - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 08 2002
%F A000792 a(n)=3^(n-2-2*floor((n-1)/3))*2^(2-(n-1)mod 3) for n>1. - Hieronymus 
               Fischer (Hieronymus.Fischer(AT)gmx.de), Nov 11 2007
%F A000792 a(n)=3^floor(n/3)/(1-(n mod 3)/4), n>1 [From Kiyoshi Akima (k_akima(AT)hotmail.com), 
               Aug 31 2009]
%F A000792 a(n)=3^(floor((n-2)/3))*(2+((n-2) mod 3))), n>1 [From Kiyoshi Akima (k_akima(AT)hotmail.com), 
               Aug 31 2009]
%e A000792 a{8} = 18 because we have 18 = (8-5)*a(5) = 3*6 and one can verify that 
               this is the maximum.
%e A000792 a(5) = 6: the 7 partitions of 5 are (5), (4,1), (3,2),(3,1,1), (2,2,1),
               (2,1,1,1), (1,1,1,1,1) and the corresponding products are 5, 4, 6, 
               3, 4, 2 and 1; 6 is the largest.
%p A000792 A000792:=-(1+2*z+3*z**2+z**3)/(-1+3*z**3); [Conjectured (correctly) by 
               S. Plouffe in his 1992 dissertation.]
%t A000792 Table[ Max[ Union[ Apply[ Times, Partitions[ n ], 1 ] ] ], {n, 30} ]
%o A000792 (PARI) a(n)=floor(3^(n-4-(n-4)\3*2)*2^(-n%3))
%Y A000792 Cf. A000793, A009490, A034891, A062943.
%Y A000792 Cf. A007601, A062723, A069188, A087902.
%Y A000792 Cf. array A064364, rightmost (nonvanishing) numbers in row n>=2.
%Y A000792 See A056240 for the minimal numbers whose prime factors sums up to n.
%Y A000792 Sequence in context: A138857 A018130 A160993 this_sequence A018752 A018393 
               A018287
%Y A000792 Adjacent sequences: A000789 A000790 A000791 this_sequence A000793 A000794 
               A000795
%K A000792 nonn,easy,nice
%O A000792 0,3
%A A000792 N. J. A. Sloane (njas(AT)research.att.com).
%E A000792 More terms and better description from Therese Biedl (biedl(AT)uwaterloo.ca), 
               Jan 19 2000

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