0,3

Numbers of the form 3^n, 2*3^n, 4*3^n with a(0)=1 prepended.

Maximum of products of partitions of n: maximal value of Product k_i for any way of writing n = Sum k_i.

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

Also the maximum number of cliques possible in a graph with n vertices, for n >= 2 (cf. Capobianco and Molluzzo). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 15 2001

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

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

Maximal number of divisors that are possible amongst numbers m such that A080256(m)=n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Oct 13 2003

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

a(n) is largest number of complexity n in sense of A005520. - David W. Wilson (davidwwilson(AT)comcast.net), Oct 03 2005

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

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

Brian R. Barwell, Cutting String and Arranging Counters, J. Rec. Math., 4 (1971), 164-168.

B. R. Barwell, Journal of Recreational Mathematics, "Maximum Product": Solution to Prob. 2004;25(4) 1993 Baywood NY.

R. Bercov and L. Moser, On abelian permutation groups, Canad. Math. Bull., 8 (1965), 627-630.

M. Capobianco and J. C. Molluzzo, Examples and Counterexamples in Graph Theory, p. 207. North-Holland: 1978.

Tomislav Doslic, Maximum Product Over Partitions Into Distinct Parts, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.8.

S. L. Greitzer, International Mathematical Olympiads 1959-1977, Prob. 1976/4 pp. 18;182-3 NML vol. 27 MAA 1978

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 396.

P. R. Halmos, Problems for Mathematicians Young and Old, Math. Assoc. Amer., 1991, pp. 30-31 and 188.

E. F. Krause, "Maximizing The Product of Summands", Mathematics Magazine, MAA Oct 1996, Vol. 69, no. 5 pp. 270-271.

L. C. Larson, Problem-Solving Through Problems. Problem 1.1.4 pp. 7. Sprnger-Verlag 1983.

J. W. Moon and L. Moser, On cliques in graphs, Israel J. Math. 3 (1965), 23-28.

D. J. Newman, A Problem Seminar. Problem 15 pp. 5;15. Spinger-Verlag 1982.

D. A. Rawsthorne, How many 1's are needed?, Fib. Quart. 27 (1989), 14-17.

A. C.-C. Yao, On a problem of Katona on minimal separating systems, Discrete Math., 15 (1976), 193-199.

T. D. Noe, Table of n, a(n) for n=0..500

MathPro, 20000 Problems Under the Sea, Problem 14856.Putnam 1979/A1

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

J. Scholes, 40th Putnam 1979 Problem A1

J. Scholes, 18th IMO 1976 Problem 4

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

a(3n) = 3^n; a(3n+1) = 4*3^(n-1) for n>0; a(3n+2) = 2*3^n.

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)lhsystems.com), Feb 08 2002

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

a{8} = 18 because we have 18 = (8-5)*a(5) = 3*6, and one can verify that this is the maximum.

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.

A000792:=-(1+2*z+3*z**2+z**3)/(-1+3*z**3); [Conjectured by S. Plouffe in his 1992 dissertation.]

Table[ Max[ Union[ Apply[ Times, Partitions[ n ], 1 ] ] ], {n, 30} ]

(PARI) a(n)=floor(3^(n-4-(n-4)\3*2)*2^(-n%3))

Cf. A000793, A009490, A034891, A062943.

Cf. A007601, A062723, A069188, A087902.

Cf. array A064364, rightmost (nonvanishing) numbers in row n>=2.

See A056240 for the minimal numbers whose prime factors sums up to n.

Adjacent sequences: A000789 A000790 A000791 this_sequence A000793 A000794 A000795

Sequence in context: A018669 A138857 A018130 this_sequence A018752 A018393 A018287

nonn,easy,nice

njas

More terms and better description from Therese Biedl (biedl(AT)uwaterloo.ca), Jan 19 2000