A212721 -id:A212721 - 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

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

A225632 Irregular table read by rows: n-th row gives distinct values of successively iterated Landau-like functions for n, starting with the initial value 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 12, 1, 6, 30, 60, 1, 6, 30, 60, 1, 12, 84, 420, 1, 15, 120, 840, 1, 20, 180, 1260, 2520, 1, 30, 210, 840, 2520, 1, 30, 420, 4620, 13860, 27720, 1, 60, 660, 4620, 13860, 27720, 1, 60, 780, 8580, 60060, 180180, 360360

Offset: 1

Antti Karttunen, May 13 2013

The leftmost column of table (the initial term of each row, T(n,1)) is 1, corresponding to lcm(1,1,...,1) computed from the {1+1+...+1} partition of n, after which, on the same row, each further term T(n,i) is computed by finding such a partition [p1,p2,...,pk] of n so that value of lcm(T(n,i-1), p1,p2,...,pk) is maximized, until finally A003418(n) is reached, which will be listed as the last term of row n (as the result would not change after that, if we continued the same process).

			The first fifteen rows of table are:
  1;
  1,   2;
  1,   3,    6;
  1,   4,   12;
  1,   6,   30,    60;
  1,   6,   30,    60;
  1,  12,   84,   420;
  1,  15,  120,   840;
  1,  20,  180,  1260,   2520;
  1,  30,  210,   840,   2520;
  1,  30,  420,  4620,  13860,  27720;
  1,  60,  660,  4620,  13860,  27720;
  1,  60,  780,  8580,  60060, 180180, 360360;
  1,  84, 1260, 16380, 180180, 360360;
  1, 105, 4620, 60060, 180180, 360360;

Alois P. Heinz, Rows n = 1..150, flattened
Index entries for sequences related to lcm's

Cf. A225634 (length of n-th row), A000793 (n>=2 gives the second column).
Cf. A225629 (second largest/rightmost term of n-th row).
Cf. A003418 (largest/rightmost term of n-th row).
Cf. A225630, A225631, A225635, A212721.
Cf. A225642 (row n starts from n instead of 1).
Cf. A226055 (the first term common with A225642 on the n-th row).
Cf. A225638 (distance to that first common term from the beginning of the row n).
Cf. A226056 (number of trailing terms common with A225642 on the n-th row).

b:= proc(n, i) option remember; `if`(n=0, {1},
      `if`(i<1, {}, {seq(map(x->ilcm(x, `if`(j=0, 1, i)),
       b(n-i*j, i-1))[], j=0..n/i)}))
    end:
T:= proc(n) option remember; local d, h, l, ll;
      l:= b(n$2); ll:= NULL; d:=1; h:=0;
      while d<>h do ll:= ll, d; h:= d;
        d:= max(seq(ilcm(h, i), i=l))
      od; ll
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, May 29 2013

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Table[Map[Function[{x}, LCM[x, If[j==0, 1, i]]], b[n-i*j, i-1]], {j, 0, n/i}]]]; T[n_] := T[n] = Module[{d, h, l, ll}, l=b[n, n]; ll={}; d=1; h=0; While[d != h, AppendTo[ll, d]; h=d; d = Max[ Table[LCM[h, i], {i, l}]]]; ll]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

A339095 Triangle read by rows: T(n,k) is the number of partitions of n with product of parts equal to k, 1 <= k <= A000792(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 0, 3, 0, 0, 1, 3, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 0, 3, 0, 1, 1, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 1

Offset: 1

Abhimanyu Kumar, Nov 23 2020

The product of the parts of a partition is called its norm.

			For n=6 the partitions and their counts for each norm are given in the table below.
  Relevant partition(s)  | Norm | Count
  1+1+1+1+1+1+1          | 1    | 1
  2+1+1+1+1              | 2    | 1
  3+1+1+1                | 3    | 1
  4+1+1, 2+2+1+1         | 4    | 2
  5+1                    | 5    | 1
  6, 3+2+1               | 6    | 2
  4+2, 2+2+2             | 8    | 2
  3+3                    | 9    | 1
The number of partitions of 6 with norm value 4 are 2, expressed as T(6,4)=2. Similarly, T(6,7)=0 because there is no partition of 6 with norm 7.
So the 6th row is 1, 1, 1, 2, 1, 2, 0, 2, 1.
First few rows of the array are:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 1, 2;
  1, 1, 1, 2, 1, 1;
  1, 1, 1, 2, 1, 2, 0, 2, 1;
  1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 2;
  1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 0, 3, 0, 0, 1, 3, 0, 1;
  ...

Abhimanyu Kumar and Meenakshi Rana, On the treatment of partitions as factorization and further analysis, Journal of the Ramanujan Mathematical Society 35(3), 263-276 (2020).

Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019.

Cf. A000041 (row sums), A000792 (row lengths), A001055, A118851, A212721.

row(n) = {my(list = List()); forpart(p=n, listput(list, vecprod(Vec(p)));); my(vlist = Vec(list)); my(v = vector(vecmax(vlist))); for (i=1, #vlist, v[vlist[i]]++); v;} \\ Michel Marcus, Nov 26 2020

Let the number of partitions of n having the norm value k refer to the norm counting function T(n,k). The following properties hold true:
Max_{n=1..oo} T(n,k) = A001055(k).
Sum_{k=1..A000792(n)} T(n,k) = A000041(n).
Sum_{n>=1} T(n+1,k) - T(n,k) = A001055(k) - 1.
G.f.: Sum_{n>=0} Sum_{k>=1} ((q^n)/(k^s))*T(n,k) = Product_{m>=1}(1-((q^m)/(m^s)))^(-1).
n*Sum_{k=1..A000792(n)} T(n,k) = Sum_{m=1..n} (Sum_{k=1..A000792(n-m)} T(n-m,i)*k^(-s))*(Sum_{d|m} (d/m)^(d*s-1))

More terms from Michel Marcus, Nov 26 2020

cp's OEIS Frontend

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

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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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A225632 Irregular table read by rows: n-th row gives distinct values of successively iterated Landau-like functions for n, starting with the initial value 1.

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A339095 Triangle read by rows: T(n,k) is the number of partitions of n with product of parts equal to k, 1 <= k <= A000792(n).

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A140436 a(n) is the maximum number of partitions of n with the same product.

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