A009490 -id:A009490 - cp's OEIS frontend

A000792 a(n) = max{(n - i)*a(i) : i < n}; a(0) = 1.

1, 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 243, 324, 486, 729, 972, 1458, 2187, 2916, 4374, 6561, 8748, 13122, 19683, 26244, 39366, 59049, 78732, 118098, 177147, 236196, 354294, 531441, 708588, 1062882, 1594323, 2125764, 3188646, 4782969, 6377292

Offset: 0

N. J. A. Sloane

Numbers of the form 3^k, 2*3^k, 4*3^k with a(0) = 1 prepended.
If a set of positive numbers has sum n, this is the largest value of their product.
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 two 2's (see formula lines below).
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. [Bercov and Moser] - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
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]
Every triple of alternate terms {3*k, 3*k+2, 3*k+4} in the sequence forms a geometric progression with first term 3^k and common ratio 2. - Lekraj Beedassy, 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, Apr 24 2003
Maximal number of divisors that are possible among numbers m such that A080256(m) = n. - Lekraj Beedassy, Oct 13 2003
Or, numbers of the 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, Jan 07 2005
a(n) is the largest number of complexity n in the sense of A005520 (A005245). - David W. Wilson, 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, Apr 29 2002
a(n) for n >= 1 is a paradigm shift sequence with procedural length p = 0, in the sense of A193455. - Jonathan T. Rowell, Jul 26 2011
a(n) = largest term of n-th row in A212721. - Reinhard Zumkeller, Jun 14 2012
For n >= 2, a(n) is the largest number whose prime divisors (with multiplicity) add to n, whereas the smallest such number (resp. smallest composite number) is A056240(n) (resp. A288814(n)). - David James Sycamore, Nov 23 2017
For n >= 3, a(n+1) = a(n)*(1 + 1/s), where s is the smallest prime factor of a(n). - David James Sycamore, Apr 10 2018

			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.
G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 18*x^8 + ...

B. 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.
M. Capobianco and J. C. Molluzzo, Examples and Counterexamples in Graph Theory, p. 207. North-Holland: 1978.
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.
L. C. Larson, Problem-Solving Through Problems. Problem 1.1.4 pp. 7. Springer-Verlag 1983.
D. J. Newman, A Problem Seminar. Problem 15 pp. 5;15. Springer-Verlag 1982.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
R. Bercov and L. Moser, On Abelian permutation groups, Canad. Math. Bull. 8 (1965) 627-630.
Walter Bridges and William Craig, On the distribution of the norm of partitions, arXiv:2308.00123 [math.CO], 2023.
J. Arias de Reyna and J. van de Lune, The question "How many 1's are needed?" revisited, arXiv preprint arXiv:1404.1850 [math.NT], 2014. See M_n.
J. Arias de Reyna and J. van de Lune, Algorithms for determining integer complexity, arXiv preprint arXiv:1404.2183 [math.NT], 2014.
Nigel Derby, 96.21 The MaxProduct partition, The Mathematical Gazette 96:535 (2012), pp. 148-151.
Tomislav Doslic, Maximum Product Over Partitions Into Distinct Parts, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.8.
Hans Havermann, Tables of sum-of-prime-factors sequences (overview with links to the first 50000 sums).
J. Iraids, K. Balodis, J. Cernenoks, M. Opmanis, R. Opmanis and K. Podnieks, Integer Complexity: Experimental and Analytical Results, arXiv preprint arXiv:1203.6462 [math.NT], 2012.
Andrew Kenney and Caroline Shapcott, Maximum Part-Products of Odd Palindromic Compositions, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.6.
E. F. Krause, Maximizing The Product of Summands, Mathematics Magazine, MAA Oct 1996, Vol. 69, no. 5 pp. 270-271.
MathPro, 20000 Problems Under the Sea, Problem 14856.Putnam 1979/A1.
J. W. Moon and L. Moser, On cliques in graphs, Israel J. Math. 3 (1965), 23-28.
Natasha Morrison and Alex Scott, Maximizing the number of induced cycles in a graph, Preprint, 2016. See f_2(n).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
F. Pluvinage, Developing problem solving experiences in practical action projects, The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos. 1 & 2, pp. 219-244.
D. A. Rawsthorne, How many 1's are needed?, Fib. Quart. 27 (1989), 14-17.
J. T. Rowell, Solution Sequences for the Keyboard Problem and its Generalizations, Journal of Integer Sequences, 18 (2015), #15.10.7.
Robert Schneider and Andrew V. Sills, The Product of Parts or "Norm" of a Partition, Integers (2020) Vol. 20A, Article #A13.
J. Scholes, 40th Putnam 1979 Problem A1.
J. Scholes, 18th IMO 1976 Problem 4.
Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019.
V. Vatter, Maximal independent sets and separating covers, Amer. Math. Monthly, 118 (2011), 418-423.
Robert G. Wilson v, Letter to N. J. Sloane, circa 1991.
A. C.-C. Yao, On a problem of Katona on minimal separating systems, Discrete Math., 15 (1976), 193-199.
Index to sequences related to the complexity of n
Index entries for linear recurrences with constant coefficients, signature (0,0,3).

See A007600 for a left inverse.
Cf. A000793, A009490, A034891, A062943, A007601, A062723, A069188, A087902.
Cf. array A064364, rightmost (nonvanishing) numbers in row n >= 2.
See A056240 and A288814 for the minimal numbers whose prime factors sums up to n.
A000792, A178715, A193286, A193455, A193456, and A193457 are closely related as paradigm shift sequences for (p = 0, ..., 5 respectively).
Cf. A202337 (subsequence).
Cf. A038754, A005245, A005520.

a000792 n = a000792_list !! n
a000792_list = 1 : f [1] where
   f xs = y : f (y:xs) where y = maximum $ zipWith (*) [1..] xs
-- Reinhard Zumkeller, Dec 17 2011

I:=[1,1,2,3,4]; [n le 5 select I[n] else 3*Self(n-3): n in [1..45]]; // Vincenzo Librandi, Apr 14 2015

A000792 := proc(n)
    m := floor(n/3) ;
    if n mod 3 = 0 then
        3^m ;
    elif n mod 3 = 1 then
        4*3^(m-1) ;
    else
        2*3^m ;
    end if;
    floor(%) ;
end proc: # R. J. Mathar, May 26 2013

a[1] = 1; a[n_] := 4* 3^(1/3 *(n - 1) - 1) /; (Mod[n, 3] == 1 && n > 1); a[n_] := 2*3^(1/3*(n - 2)) /; Mod[n, 3] == 2; a[n_] := 3^(n/3) /; Mod[n, 3] == 0; Table[a[n], {n, 0, 40}]
CoefficientList[Series[(1 + x + 2x^2 + x^4)/(1 - 3x^3), {x, 0, 50}], x] (* Harvey P. Dale, May 01 2011 *)
f[n_] := Max[ Times @@@ IntegerPartitions[n, All, Prime@ Range@ PrimePi@ n]]; f[1] = 1; Array[f, 43, 0] (* Robert G. Wilson v, Jul 31 2012 *)
a[ n_] := If[ n < 2, Boole[ n > -1], 2^Mod[-n, 3] 3^(Quotient[ n - 1, 3] + Mod[n - 1, 3] - 1)]; (* Michael Somos, Jan 23 2014 *)
Join[{1, 1}, LinearRecurrence[{0, 0, 3}, {2, 3, 4}, 50]] (* Jean-François Alcover, Jan 08 2019 *)
Join[{1,1},NestList[#+Divisors[#][[-2]]&,2,41]] (* James C. McMahon, Aug 09 2024 *)

{a(n) = floor( 3^(n - 4 - (n - 4) \ 3 * 2) * 2^( -n%3))}; /* Michael Somos, Jul 23 2002 */

lista(nn) = {print1("1, 1, "); print1(a=2, ", "); for (n=1, nn, a += a/divisors(a)[2]; print1(a, ", "););} \\ Michel Marcus, Apr 14 2015

A000792(n)=if(n>1,3^((n-2)\3)*(2+(n-2)%3),1) \\ M. F. Hasler, Jan 19 2019

G.f.: (1 + x + 2*x^2 + x^4)/(1 - 3*x^3). - Simon Plouffe in his 1992 dissertation.
a(3n) = 3^n; a(3*n+1) = 4*3^(n-1) for n > 0; a(3*n+2) = 2*3^n.
a(n) = 3*a(n-3) if n > 4. - Henry Bottomley, Nov 29 2001
a(n) = n if n <= 2, otherwise a(n-1) + Max{gcd(a(i), a(j)) | 0 < i < j < n}. - Reinhard Zumkeller, Feb 08 2002
A007600(a(n)) = n; Andrew Chi-Chih Yao attributes this observation to D. E. Muller. - Vincent Vatter, Apr 24 2006
a(n) = 3^(n - 2 - 2*floor((n - 1)/3))*2^(2 - (n - 1) mod 3) for n > 1. - Hieronymus Fischer, Nov 11 2007
From Kiyoshi Akima (k_akima(AT)hotmail.com), Aug 31 2009: (Start)
a(n) = 3^floor(n/3)/(1 - (n mod 3)/4), n > 1.
a(n) = 3^(floor((n - 2)/3))*(2 + ((n - 2) mod 3)), n > 1. (End)
a(n) = (2^b)*3^(C - (b + d))*(4^d), n > 1, where C = floor((n + 1)/3), b = max(0, ((n + 1) mod 3) - 1), d = max(0, 1 - ((n + 1) mod 3)). - Jonathan T. Rowell, Jul 26 2011
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 - x / (1 + x / (1 + x^2 / (1 + x))))))). - Michael Somos, May 12 2012
3*a(n) = 2*a(n+1) if n > 1 and n is not divisible by 3. - Michael Somos, Jan 23 2014
a(n) = a(n-1) + largest proper divisor of a(n-1), n > 2. - Ivan Neretin, Apr 13 2015
a(n) = max{a(i)*a(n-i) : 0 < i < n} for n >= 4. - Jianing Song, Feb 15 2020
a(n+1) = a(n) + A038754(floor( (2*(n-1) + 1)/3 )), for n > 1. - Thomas Scheuerle, Oct 27 2022

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

A000793 Landau's function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 12, 15, 20, 30, 30, 60, 60, 84, 105, 140, 210, 210, 420, 420, 420, 420, 840, 840, 1260, 1260, 1540, 2310, 2520, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 13860, 16380, 16380, 27720, 30030, 32760, 60060, 60060, 60060, 60060, 120120

Offset: 0

N. J. A. Sloane

Also the largest orbit size (cycle length) for the permutation A057511 acting on Catalan objects (e.g., planar rooted trees, parenthesizations). - Antti Karttunen, Sep 07 2000
Grantham mentions that he computed a(n) for n <= 500000.
An easy lower bound is a(n) >= A002110(max{ m | A007504(m) <= n}), with strict inequality if n is not in A007504 (sum of the first m primes). Indeed, if A007504(m) <= n, the partition of n into the first m primes and maybe one additional term will have an LCM greater than or equal to primorial(m). If n > A007504(m) then a(n) >= (3/2)*A002110(m) by replacing the initial 2 by 3. But even for n = A007504(m), one has a(n) > A002110(m) for m > 8, since replacing 2+23 in 2+3+5+7+11+13+17+19+23 by 16+9, one has an LCM of 8*3*primorial(8) > primorial(9) because 24 > 23. - M. F. Hasler, Mar 29 2015
Maximum degree of the splitting field of a polynomial of degree n over a finite field, since over a finite field the degree of the splitting field is the least common multiple of the degrees of the irreducible polynomial factors of the polynomial. - Charles R Greathouse IV, Apr 27 2015
Maximum order of the elements in the symmetric group S_n. - Jianing Song, Dec 12 2021

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 6*x^6 + 12*x^7 + 15*x^8 + ...
From _Joerg Arndt_, Feb 15 2013: (Start)
The 15 partitions of 7 are the following:
[ #]  [ partition ]   lcm( parts )
[ 1]  [ 1 1 1 1 1 1 1 ]   1
[ 2]  [ 1 1 1 1 1 2 ]   2
[ 3]  [ 1 1 1 1 3 ]   3
[ 4]  [ 1 1 1 2 2 ]   2
[ 5]  [ 1 1 1 4 ]   4
[ 6]  [ 1 1 2 3 ]   6
[ 7]  [ 1 1 5 ]   5
[ 8]  [ 1 2 2 2 ]   2
[ 9]  [ 1 2 4 ]   4
[10]  [ 1 3 3 ]   3
[11]  [ 1 6 ]   6
[12]  [ 2 2 3 ]   6
[13]  [ 2 5 ]  10
[14]  [ 3 4 ]  12  (max)
[15]  [ 7 ]   7
The maximum (LCM) value attained is 12, so a(7) = 12.
(End)

J. Haack, "The Mathematics of Steve Reich's Clapping Music," in Bridges: Mathematical Connections in Art, Music and Science: Conference Proceedings, 1998, Reza Sarhangi (ed.), 87-92.
Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, p. 223.
J.-L. Nicolas, On Landau's function g(n), pp. 228-240 of R. L. Graham et al., eds., Mathematics of Paul Erdős I.
S. M. Shah, An inequality for the arithmetical function g(x), J. Indian Math. Soc., 3 (1939), 316-318. [See below for a scan of the first page.]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 0..814 from David Wasserman)
Giedrius Alkauskas, Colouring monohedral tilings: defects and grain boundaries, 2024.
Joerg Arndt, Table of n, a(n) for n = 0..65536 (xz compressed).
Jan Brandts and A. Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
Marc Deléglise, Jean-Louis Nicolas, and Paul Zimmermann, Landau's function for one million billions, arXiv:0803.2160 [math.NT], 2008.
Marc Deléglise and Jean-Louis Nicolas, On the Largest Product of Primes with Bounded Sum, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.8.
Marc Deléglise and Jean-Louis Nicolas, The Landau function and the Riemann hypothesis, Univ. Lyon (France, 2019).
John D. Dixon and Daniel Panario, The Degree of the Splitting Field of a Random Polynomial over a Finite Field, The Electronic Journal of Combinatorics 11:1 (2004).
FindStat - Combinatorial Statistic Finder, The order of a permutation
Jon Grantham, The largest prime dividing the maximal order of an element of S_n, Math. Comput. 64, No. 209, 407-410 (1995).
Joel K. Haack, The Mathematics of Steve Reich's Clapping Music,in Bridges: Mathematical Connections in Art, Music and Science: Conference Proceedings, 1998, Reza Sarhangi (ed.), 87-92.
Jan Elmar Krauskopf, Andreas Rauh, and Andreas Hein, Discrete simulation of maypole braiding machines to create collision-free braiding programmes, Heliyon (2025), Art. No. e42917.
J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186.
Jean-Pierre Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique, (French) [Explicit upper bound on the maximum order of an element of the symmetric group] Ann. Fac. Sci. Toulouse Math. (5) 6 (1984), no. 3-4, 269--281 (1985). MR0799599 (87a:11093).
Jean-Pierre Massias, Jean-Louis Nicolas, and Guy Robin, Effective bounds for the maximal order of an element in the symmetric group, Math. Comp. 53 (1989), no. 188, 665--678. MR0979940 (90e:11139).
W. Miller, The Maximum Order of an Element of Finite Symmetric Group, Am. Math. Monthly, Jun-Jul 1987, pp. 497-506.
Jean-Louis Nicolas, Sur l'ordre maximum d'un élément dans le groupe Sn des permutations, Acta Arith. 14, 315-332 (1968).
Jean-Louis Nicolas, Ordre maximal d'un élément du groupe S_n des permutations et 'highly composite numbers', Bull. Soc. Math. France 97 (1969), 129-191.
Jean-Louis Nicolas, Calcul de l'ordre maximum d'un élément du groupe symétrique S_n Revue française d'informatique et de recherche opérationnelle, série rouge 3.2 (1969): 43-50.
Jean-Louis Nicolas, Ordre maximal d'un e'le'ment du'un groupe de permutations, C. R. Acad. Sci. Paris, A, 270 (1970), 1-4. [Annotated scanned copy]
Roger D. Nussbaum, Lunel Verduyn, and M. Sjoerd, Asymptotic estimates for the periods of periodic points of non-expansive maps, Ergodic Theory Dynam. Systems 23 (2003), no. 4, pp. 1199-1226. MR1997973 (2004m:37033).
S. M. Shah, An inequality for the arithmetical function g(x) (scan of first page).
A. Wechsler, Re: Question (A000793(A007504(n)) =? A002110(n)), SeqFan mailing list, Mar 29 2015.
Eric Weisstein's World of Mathematics, Landau's Function
Herbert S. Wilf, The asymptotics of e^P(z) and the number of elements of each order in S_n, Bull. Amer. Math. Soc., 15.2 (1986), 225-232.
Index entries for sequences related to lcm's
Index entries for "core" sequences

Cf. A000792, A009490, A034891, A057731, A074859, A128305, A129759, A225655, A225648-A225650, A225651, A225636, A225558.

Row 1 of A225630.

a000793 = maximum . map (foldl lcm 1) . partitions where
   partitions n = ps 1 n where
      ps x 0 = [[]]
      ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Mar 29 2015

A000793 := proc(n)
    l := 1:
    p := combinat[partition](n):
    for i from 1 to combinat[numbpart](n) do
        if ilcm( p[i][j] $ j=1..nops(p[i])) > l then
            l := ilcm( p[i][j] $ j=1..nops(p[i]))
        end if:
    end do:
    l ;
end proc:
seq(A000793(n),n=0..30) ; # James Sellers, Dec 07 2000
seq( max( op( map( x->ilcm(op(x)), combinat[partition](n)))), n=0..30); # David Radcliffe, Feb 28 2006
# third Maple program:
b:= proc(n, i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, max(b(n, i-1),
           seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))))
    end:
a:=n->b(n, `if`(n<8, 3, numtheory[pi](ceil(1.328*isqrt(n*ilog(n)))))):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 16 2013

f[n_] := Max@ Apply[LCM, IntegerPartitions@ n, 1]; Array[f, 47] (* Robert G. Wilson v, Oct 23 2011 *)
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0 || i<1, 1, Max[b[n, i-1], Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n] // Floor}]]]]; a[n_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 07 2014, after Alois P. Heinz *)

{a(n) = my(m, t, j, u); if( n<2, n>=0, m = ceil(n / exp(1)); t = ceil( (n/m)^m ); j=1; for( i=2, t, u = factor(i); u = sum( k=1, matsize(u)[1], u[k,1]^u[k,2]); if( u<=n, j=i)); j)}; /* Michael Somos, Oct 20 2004 */

c=0;A793=apply(t->eval(concat(Vec(t)[#Str(c++) .. -1])),select(t->#t,readstr("/tmp/b000793.txt")));A000793(n)=A793[n+1] \\ Assumes the b-file in the /tmp (or C:\tmp) folder. - M. F. Hasler, Mar 29 2015

A008475(n)=my(f=factor(n)); sum(i=1,#f~,f[i,1]^f[i,2]);
a(n)=
{
  if(n<2, return(1));
  forstep(i=ceil(exp(1.05315*sqrt(log(n)*n))), 2, -1,
    if(A008475(i)<=n, return(i))
  );
  1;
} \\ Charles R Greathouse IV, Apr 28 2015

{ \\ translated from code given by Tomas Rokicki
  my( N = 100 );
  my( V = vector(N,j,1) );
   forprime (i=2, N,  \\ primes i
      forstep (j=N, i,  -1,
         my( hi = V[j] );
         my( pp = i );  \\ powers of prime i
         while ( pp<=j,  \\ V[] is 1-based
             hi = max(if(j==pp, pp, V[j-pp]*pp), hi);
             pp *= i;
         );
         V[j] = hi;
      );
   );
   print( V );  \\ all values
\\   print( V[N] );  \\ just a(N)
\\  print("0 1");  for (n=1, N, print(n, " ", V[n]) );  \\ b-file
} \\ Joerg Arndt, Nov 14 2016

{a(n) = my(m=1); if( n<0, 0, forpart(v=n, m = max(m, lcm(Vec(v)))); m)}; /* Michael Somos, Sep 04 2017 */

from sympy import primerange
def aupton(N): # compute terms a(0)..a(N)
    V = [1 for j in range(N+1)]
    for i in primerange(2, N+1):
        for j in range(N, i-1, -1):
            hi = V[j]
            pp = i
            while pp <= j:
                hi = max((pp if j==pp else V[j-pp]*pp), hi)
                pp *= i
            V[j] = hi
    return V
print(aupton(47)) # Michael S. Branicky, Oct 09 2022 after Joerg Arndt

from sympy import primerange,sqrt,log,Rational
def f(N): # compute terms a(0)..a(N)
    V = [1 for j in range(N+1)]
    if N < 4:
        C = 2
    else:
        C = Rational(166,125)
    for i in primerange(C*sqrt(N*log(N))):
        for j in range(N, i-1, -1):
            hi = V[j]
            pp = i
            while pp <= j:
                hi = max(V[j-pp]*pp, hi)
                pp *= i
            V[j] = hi
    return V
# Philip Turecek, Mar 31 2023

def a(n):
  return max([lcm(l) for l in Partitions(n)])
# Philip Turecek, Mar 28 2023

;; A naive algorithm searching through all partitions of n:
(define (A000793 n) (let ((maxlcm (list 0))) (fold_over_partitions_of n 1 lcm (lambda (p) (set-car! maxlcm (max (car maxlcm) p)))) (car maxlcm)))
(define (fold_over_partitions_of m initval addpartfun colfun) (let recurse ((m m) (b m) (n 0) (partition initval)) (cond ((zero? m) (colfun partition)) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (addpartfun i partition)) (if (< i (min b m)) (loop (+ 1 i))))))))
;; From Antti Karttunen, May 17 2013.

Landau: lim_{n->oo} (log a(n)) / sqrt(n log n) = 1.
For bounds, see the Shah and Massias references.
For n >= 2, a(n) = max_{k} A008475(k) <= n. - Joerg Arndt, Nov 13 2016

More terms from David W. Wilson

Removed erroneous comment about a(16) which probably originated from misreading a(15)=105 as a(16) because of offset=0: a(16) = 4*5*7 = 140 is correct as it stands. - M. F. Hasler, Feb 02 2009

A034891 Number of different products of partitions of n; number of partitions of n into prime parts (1 included); number of distinct orders of Abelian subgroups of symmetric group S_n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 36, 45, 55, 67, 81, 98, 117, 140, 166, 196, 231, 271, 317, 369, 429, 496, 573, 660, 758, 869, 993, 1133, 1290, 1465, 1662, 1881, 2125, 2397, 2699, 3035, 3407, 3820, 4276, 4780, 5337, 5951, 6628, 7372, 8191, 9090

Offset: 0

Wouter Meeussen

a(n) = length of n-th row in A212721. - Reinhard Zumkeller, Jun 14 2012

Number of partitions of n into noncomposite parts. - Omar E. Pol, Jun 23 2022

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..1000 from T. D. Noe)
Index entries for sequences related to partitions

Cf. A000041, A000792, A000793, A009490, A008578, A140436, A353188.

a034891 = length . a212721_row  -- Reinhard Zumkeller, Jun 14 2012

b:= proc(n, i) option remember; `if`(n=0, 1, (p->
      `if`(i<0, 0, b(n, i-1)+ `if`(p>n, 0,
         b(n-p, i))))(`if`(i<1, 1, ithprime(i))))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 15 2013

Table[ Length[ Union[ Apply[ Times, Partitions[ n], 1]]], {n, 30}]
CoefficientList[ Series[ (1/(1 - x)) Product[1/(1 - x^Prime[i]), {i, 100}], {x, 0, 50}], x] (* Robert G. Wilson v, Aug 17 2013 *)
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0, 1, If[i<0, 0, b[n, i-1] + If[p>n, 0, b[n-p, i]]]]]; a[n_] := b[n, PrimePi[n] ]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)

[Partitions(n, parts_in=(prime_range(n+1)+[1])).cardinality() for n in xsrange(1000)] # Giuseppe Coppoletta, Jul 11 2016

G.f.: (1/(1-x))*(1/Product_{k>0} (1-x^prime(k))). a(n) = (1/n)*Sum_{k=1..n} A074372(k)*a(n-k). Partial sums of A000607. - Vladeta Jovovic, Sep 19 2002

a(n) = A000041(n) - A353188(n). - Omar E. Pol, Jun 23 2022

More terms from Vladeta Jovovic

a(0)=1 from Michael Somos, Feb 05 2011

A023893 Number of partitions of n into prime power parts (1 included); number of nonisomorphic Abelian subgroups of symmetric group S_n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 105, 134, 171, 215, 269, 335, 415, 511, 626, 764, 929, 1125, 1356, 1631, 1953, 2333, 2776, 3296, 3903, 4608, 5427, 6377, 7476, 8744, 10205, 11886, 13818, 16032, 18565, 21463, 24768, 28536

Offset: 0

Olivier Gérard

nonn

			From _Gus Wiseman_, Jul 28 2022: (Start)
The a(0) = 1 through a(6) = 10 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (33)
           (11)  (21)   (22)    (32)     (42)
                 (111)  (31)    (41)     (51)
                        (211)   (221)    (222)
                        (1111)  (311)    (321)
                                (2111)   (411)
                                (11111)  (2211)
                                         (3111)
                                         (21111)
                                         (111111)
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to groups

Cf. A009490, A023894 (first differences), A062297 (number of Abelian subgroups).
Cf. A018819, A062051, A131995.
The multiplicative version (factorizations) is A000688.
Not allowing 1's gives A023894, strict A054685, ranked by A355743.
The version for just primes (not prime-powers) is A034891, strict A036497.
The strict version is A106244.
These partitions are ranked by A302492.
A000041 counts partitions, strict A000009.
A001222 counts prime-power divisors.
A072233 counts partitions by sum and length.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
Cf. A001970, A055887, A063834, A085970, A279784, A295935, A356065.

Table[Length[Select[IntegerPartitions[n],Count[Map[Length,FactorInteger[#]], 1] == Length[#] &]], {n, 0, 35}] (* Geoffrey Critzer, Oct 25 2015 *)
nmax = 50; Clear[P]; P[m_] := P[m] = Product[Product[1/(1-x^(p^k)), {k, 1, m}], {p, Prime[Range[PrimePi[nmax]]]}]/(1-x)+O[x]^nmax // CoefficientList[ #, x]&; P[1]; P[m=2]; While[P[m] != P[m-1], m++]; P[m] (* Jean-François Alcover, Aug 31 2016 *)

lista(m) = {x = t + t*O(t^m); gf = prod(k=1, m, if (isprimepower(k), 1/(1-x^k), 1))/(1-x); for (n=0, m, print1(polcoeff(gf, n, t), ", "));} \\ Michel Marcus, Mar 09 2013

from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A023893(n):
    @lru_cache(maxsize=None)
    def c(n): return sum((p**(e+1)-p)//(p-1) for p,e in factorint(n).items())+1
    return (c(n)+sum(c(k)*A023893(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024

G.f.: (Product_{p prime} Product_{k>=1} 1/(1-x^(p^k))) / (1-x).

A222029 Triangle of number of functions in a size n set for which the sequence of composition powers ends in a length k cycle.

Original entry on oeis.org

1, 1, 3, 1, 16, 9, 2, 125, 93, 32, 6, 1296, 1155, 480, 150, 24, 20, 16807, 17025, 7880, 3240, 864, 840, 262144, 292383, 145320, 71610, 24192, 26250, 720, 0, 0, 504, 0, 420, 4782969, 5752131, 3009888, 1692180, 653184, 773920, 46080, 5040, 0, 32256, 0, 26880, 0, 0, 2688

Offset: 0

Chad Brewbaker, May 14 2013

If you take the powers of a finite function you generate a lollipop graph. This table organizes the lollipops by cycle size. The table organized by total lollipop size with the tail included is A225725.

Warning: For T(n,k) after the sixth row there are zero entries and k can be greater than n: T(7,k) = |{1=>262144, 2=>292383, 3=>145320, 4=>71610, 5=>24192, 6=>26250, 7=>720, 8=>0, 9=>0, 10=>504, 11=>0, 12=>420}|.

			T(1,1) = |{[0]}|, T(2,1) = |{[0,0],[0,1],[1,1]}|, T(2,2) = |{[0,1]}|.
Triangle starts:
       1;
       1;
       3,      1;
      16,      9,      2;
     125,     93,     32,     6;
    1296,   1155,    480,   150,    24,    20;
   16807,  17025,   7880,  3240,   864,   840;
  262144, 292383, 145320, 71610, 24192, 26250, 720, 0, 0, 504, 0, 420;
  ...

Alois P. Heinz, Rows n = 0..30, flattened
Chad Brewbaker, Ruby program for A222029

Columns k=1-10 give A000272, A163951, A163952, A291110, A291111, A291112, A291113, A291114, A291115, A291116.
Rows sums give A000312.
Row lengths are A000793.
Number of nonzero elements of rows give A009490.
Last elements of rows give A162682.
Main diagonal gives A290961.
Cf. A057731 (the same for permutations), A290932.

b:= proc(n, m) option remember; `if`(n=0, x^m, add((j-1)!*
      b(n-j, ilcm(m, j))*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(add(
         b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 14 2017

b[n_, m_]:=b[n, m]=If[n==0, x^m, Sum[(j - 1)!*b[n - j, LCM[m, j]] Binomial[n - 1, j - 1], {j, n}]]; T[n_]:=If[n==0, {1}, Drop[CoefficientList[Sum[b[j, 1]n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}], x], 1]]; Table[T[n], {n, 0, 10}]//Flatten (* Indranil Ghosh, Aug 17 2017 *)

from sympy.core.cache import cacheit
from sympy import binomial, Symbol, lcm, factorial as f, Poly, flatten
x=Symbol('x')
@cacheit
def b(n, m): return x**m if n==0 else sum([f(j - 1)*b(n - j, lcm(m, j))*binomial(n - 1, j - 1) for j in range(1, n + 1)])
def T(n): return Poly(sum([b(j, 1)*n**(n - j)*binomial(n - 1, j - 1) for j in range(n + 1)]),x).all_coeffs()[::-1][1:]
print([T(n) for n in range(11)]) # Indranil Ghosh, Aug 17 2017

Sum_{k=1..A000793(n)} k * T(n,k) = A290932. - Alois P. Heinz, Aug 14 2017

T(0,1)=1 prepended by Alois P. Heinz, Aug 14 2017

A051613 a(n) = partitions of n into powers of distinct primes (1 not considered a power).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 0, 3, 2, 3, 2, 4, 3, 4, 4, 4, 8, 4, 8, 6, 9, 8, 10, 10, 13, 12, 13, 16, 16, 19, 17, 21, 23, 23, 25, 29, 31, 31, 31, 37, 40, 42, 44, 48, 49, 54, 55, 64, 67, 68, 70, 77, 84, 90, 92, 99, 102, 108, 115, 127, 133, 135, 138, 150, 165, 171, 183, 186, 198, 201, 220

Offset: 0

Vladeta Jovovic

			a(16) = 8 because we can write 16 = 2^4 = 3+13 = 5+11 = 3^2+7 = 2+3+11 = 2+3^2+5 = 2^3+3+5 = 2^2+5+7.

T. D. Noe, Table of n, a(n) for n = 0..1000
J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Cf. A023894, A009490, A054685, A008475, A106245, A000961, A025473, A106244.

import Data.MemoCombinators (memo3, integral)
a051613' = p 1 2 where
   p x _ 0 = 1
   p x k m | m < qq       = 0
           | mod x q == 0 = p x (k + 1) m
           | otherwise    = p (q * x) (k + 1) (m - qq) + p x (k + 1) m
           where q = a025473 k; qq = a000961 k
-- Reinhard Zumkeller, Nov 23 2015

b:= proc(n,i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0, 1, `if`(i<1, 0, b(n,i-1)+
      add(b(n-p^j, i-1), j=1..ilog[p](n))))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 15 2013

max = 70; f[x_] := Product[ 1 + Sum[x^(Prime[n]^k), {k, 1, If[n > 4, 1, 6]}], {n, 1, PrimePi[max]}]; CoefficientList[ Series[f[x], {x, 0, max}] , x](* Jean-François Alcover, Sep 12 2012 *)

first(n)=my(x='x,pr=O(x^(n+1))+1); forprime(p=sqrtint(n)+1,n, pr*=1+x^p); forprime(p=2,sqrtint(n), pr*=1+sum(e=1,logint(n,2), x^p^e)); Vec(pr) \\ Charles R Greathouse IV, Jun 25 2017

a(n) = number of m such that A008475(m) = n.

G.f.: Product_{p prime} (1 + Sum_{k >= 1} x^(p^k)).

Better description from David W. Wilson, Apr 19 2000

A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 0, 0, 1, 0, 1, 1, 4, 2, 4, 1, 5, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 0, 4, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 3, 6, 2

Offset: 0

R. J. Mathar, Mar 18 2015

			The 5 partitions of n=4 are 1+1+1+1 (lcm=1), 1+1+2 (lcm=2), 2+2 (lcm=2), 1+3 (lcm=3) and 4 (lcm=4). So k=1, 3 and 4 appear once, k=2 appears twice.
The triangle starts:
  1 ;
  1 ;
  1  1;
  1  1  1;
  1  2  1  1;
  1  2  1  1  1  1;
  1  3  2  2  1  2;
  1  3  2  2  1  3  1  0  0  1  0  1;
  ...

Alois P. Heinz, Rows n = 0..30, flattened

Cf. A000041 (row sums), A000793 (row lengths), A213952, A074761 (diagonal), A074752 (6th column), A008642 (4th column), A002266 (5th column), A002264 (3rd column), A132270 (7th column), A008643 (8th column), A008649 (9th column), A258470 (10th column).

Cf. A009490 (number of nonzero terms of rows), A074064 (last elements of rows), A168532 (the same for gcd), A181844 (Sum k*T(n,k)).

A256067 := proc(n,k)
        local a,p ;
        a := 0 ;
        for p in combinat[partition](n) do
                ilcm(op(p)) ;
                if % = k then
                        a := a+1 ;
                end if;
        end do:
        a;
end proc:
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
      t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 27 2015

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i-1] + Function[{p}, Sum[ Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n-i*j, i-1], {j, 1, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 22 2015, after Alois P. Heinz *)

T(0,1)=1 prepended by Alois P. Heinz, Mar 27 2015

cp's OEIS Frontend

A256554 Number T(n,k) of cycle types of degree-n permutations having the k-th smallest possible order; triangle T(n,k), n>=0, 1<=k<=A009490(n), read by rows.

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A256553 Triangle T(n,k) in which the n-th row contains the increasing list of distinct orders of degree-n permutations; n>=0, 1<=k<=A009490(n).

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A034890 Duplicate of A009490.

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A000792 a(n) = max{(n - i)*a(i) : i < n}; a(0) = 1.

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A000793 Landau's function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n.

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A034891 Number of different products of partitions of n; number of partitions of n into prime parts (1 included); number of distinct orders of Abelian subgroups of symmetric group S_n.

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