A057511 -id:A057511 - cp's OEIS frontend

A079219 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=0, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the three-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

1, 1, 1, 2, 0, 1, 3, 1, 3, 1, 8, 0, 9, 0, 1, 18, 2, 22, 0, 0, 1, 43, 0, 60, 0, 0, 0, 1, 104, 5, 159, 1, 0, 3, 0, 1, 273, 0, 428, 0, 0, 0, 0, 0, 1, 702, 14, 1143, 0, 1, 9, 0, 0, 0, 1, 1870, 0, 3114, 0, 0, 0, 0, 0, 0, 0, 1, 4985, 38, 8505, 2, 0, 28, 0, 0, 3, 0, 0, 1, 13562, 0, 23475, 0, 0, 0, 0

Offset: 0

Antti Karttunen Jan 03 2002

Note: the counts given here are inclusive, i.e. T(n,d) includes also the count A079217(n,d).

The row sums equal to the left edge shifted left once = A079224 = third row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079222 and A003056 and A002262.

[seq(A079219(n),n=0..119)]; A079219 := n -> PFixedByA057511(A003056(n)+1,3, A002262(n)+1);

A079226 Number of Catalan objects fixed by five-fold application of the Catalan bijections A057511/A057512 (deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 15, 36, 108, 301, 814, 2080, 5223, 12919, 32557, 83943, 222591, 600252, 1632814, 4440240, 12043224, 32572225, 88081208, 238722759, 649725756, 1776546687, 4877740703, 13432630929, 37063472432, 102389547753

Offset: 0

Antti Karttunen Jan 03 2002

nonn

The fifth row of A079216. The leftmost edge of the triangle A079221 and also its row sums shifted by one. Occurs in A073202 as row 9259542121261050623. Cf. A057546, A079223-A079227.

Maple
```
A079226 := n -> A079216bi(n,5);
```

a(n) = A079216(n, 5)

A079221 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=0, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the five-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 0, 0, 0, 1, 6, 2, 1, 0, 5, 1, 15, 0, 0, 0, 20, 0, 1, 36, 5, 0, 1, 65, 0, 0, 1, 108, 0, 2, 0, 190, 0, 0, 0, 1, 301, 11, 0, 0, 501, 0, 0, 0, 0, 1, 814, 0, 0, 0, 1265, 0, 0, 0, 0, 0, 1, 2080, 26, 3, 2, 3105, 1, 0, 0, 0, 5, 0, 1, 5223, 0, 0, 0, 7695, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen Jan 03 2002

Note: the counts given here are inclusive, i.e. T(n,d) includes also the count A079217(n,d).

The row sums equal to the left edge shifted left once = A079226 = fifth row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079222 and A003056 and A002262.

[seq(A079221(n),n=0..119)]; A079221 := n -> PFixedByA057511(A003056(n)+1,5, A002262(n)+1);

A079224 Number of Catalan objects fixed by three-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 2, 3, 8, 18, 43, 104, 273, 702, 1870, 4985, 13562, 37038, 102266, 283774, 793189, 2227115, 6286044, 17811751, 50672898, 144639235, 414181050, 1189365940, 3424477813, 9883578364, 28589660227, 82870288432, 240672107114

Offset: 0

Antti Karttunen Jan 03 2002

nonn

The third row of A079216. The leftmost edge of the triangle A079219 and also its row sums shifted by one. Occurs in A073202 as row 43639. Cf. A057546, A079223-A079227.

Maple
```
A079224 := n -> A079216bi(n,3);
```

a(n) = A079216(n, 3)

A079225 Number of Catalan objects fixed by four-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 2, 5, 11, 30, 82, 233, 680, 2033, 6164, 18923, 58768, 184045, 581105, 1846906, 5905364, 18980465, 61292929, 198758704, 646974285, 2113163707, 6923642271, 22749608810, 74946337830, 247499313730, 819154110660, 2716779932308

Offset: 0

Antti Karttunen Jan 03 2002

nonn

The fourth row of A079216. The leftmost edge of the triangle A079220 and also its row sums shifted by one. Occurs in A073202 as row 2290625151. Cf. A057546, A079223-A079227.

Maple
```
A079225 := n -> A079216bi(n,4);
```

a(n) = A079216(n, 4)

A130919 Signature permutation of a Catalan automorphism: DEEPEN-transform of automorphism *A057511.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 11, 14, 16, 19, 10, 15, 12, 17, 18, 13, 20, 21, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 24, 29, 38, 43, 52, 26, 40, 31, 45, 54, 32, 46, 49, 50, 27, 41, 34, 48, 55, 35, 57, 58, 62, 36, 61, 59, 63, 64, 65, 67, 70, 72, 75, 79, 81

Offset: 0

Antti Karttunen, Jun 11 2007

nonn

*A130919 = DEEPEN(*A057511) = NEPEED(*A057511) = DEEPEN(DEEPEN(*A057509)) = NEPEED(NEPEED(*A057509)). See A122283, A122284 for the definitions of DEEPEN and NEPEED transforms.

Inverse: A130920. A122351(n) = A083927(A130919(A057123(n))). The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A130967 and A130968. Maximum cycle sizes seems to be given by A000793 (shifted once right).

A079220 Triangle T(n,d) (listed row-wise: T(1,1)=1, T(2,1)=1, T(2,2)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1, ...) giving the number of n-edge general plane trees with root degree d that are fixed by the four-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 0, 1, 11, 14, 0, 4, 1, 30, 36, 1, 14, 0, 1, 82, 102, 0, 48, 0, 0, 1, 233, 293, 0, 153, 0, 0, 0, 1, 680, 860, 2, 488, 0, 2, 0, 0, 1, 2033, 2575, 0, 1550, 1, 0, 0, 4, 0, 1, 6164, 7838, 0, 4920, 0, 0, 0, 0, 0, 0, 1, 18923, 24148, 5, 15672, 0, 5, 0, 14, 0, 0, 0, 1

Offset: 0

Antti Karttunen Jan 03 2002

Note: the counts given here are inclusive, i.e. T(n,d) includes also the count A079218(n,d).

The row sums equal to the left edge shifted left once = A079225 = fourth row of A079216 (the latter gives the Maple procedure PFixedByA057511). Cf. also A079217-A079222 and A003056 & A002262.

[seq(A079220(n),n=0..119)]; A079220 := n -> PFixedByA057511(A003056(n)+1,4, A002262(n)+1);

A243495 Indices in A014486 for the oriented trees that stay the same when "deep-rotated": fixed points of A057511/A057512.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 9, 15, 17, 21, 22, 23, 45, 55, 58, 63, 64, 65, 113, 124, 129, 153, 170, 185, 189, 195, 196, 197, 393, 493, 515, 524, 564, 591, 612, 617, 624, 625, 626, 1103, 1237, 1251, 1330, 1535, 1628, 1679, 1794, 1859, 1897, 1911, 1973, 2012, 2040, 2046, 2054, 2055

Offset: 0

Antti Karttunen, Jun 07 2014

nonn

Cf. A243496, A243497, A014486, A057511, A057512.

A073200 Number of simple Catalan bijections of type B.

Original entry on oeis.org

0, 1, 0, 3, 1, 0, 2, 2, 1, 0, 7, 3, 3, 1, 0, 8, 4, 2, 3, 1, 0, 6, 6, 8, 2, 3, 1, 0, 4, 5, 7, 7, 2, 3, 1, 0, 5, 7, 6, 6, 8, 2, 3, 1, 0, 17, 8, 5, 8, 7, 7, 2, 2, 1, 0, 18, 9, 4, 4, 6, 8, 7, 3, 3, 1, 0, 20, 10, 22, 5, 5, 5, 8, 4, 2, 2, 1, 0, 21, 14, 21, 17, 4, 4, 6, 5, 8, 3, 3, 1, 0

Offset: 0

Antti Karttunen, Jun 25 2002

Each row is a permutation of nonnegative integers induced by a Catalan bijection (constructed as explained below) acting on the parenthesizations/plane binary trees as encoded and ordered by A014486/A063171.
The construction process is akin to the constructive mapping of primitive recursive functions to N: we have two basic primitives, A069770 (row 0) and A072796 (row 1), of which the former swaps the left and the right subtree of a binary tree and the latter exchanges the positions of the two leftmost subtrees of plane general trees, unless the tree's degree is less than 2, in which case it just fixes it. From then on, the even rows are constructed recursively from any other Catalan bijection in this table, using one of the five allowed recursion types:
0 - Apply the given Catalan bijection and then recurse down to both subtrees of the new binary tree obtained. (last decimal digit of row number = 2)
1 - First recurse down to both subtrees of the old binary tree and only after that apply the given Catalan bijection. (last digit = 4)
2 - Apply the given Catalan bijection and then recurse down to the right subtree of the new binary tree obtained. (last digit = 6)
3 - First recurse down to the right subtree of old binary tree and only after that apply the given Catalan bijection. (last digit = 8)
4 - First recurse down to the left subtree of old binary tree, after that apply the given Catalan bijection and then recurse down to the right subtree of the new binary tree. (last digit = 0)
The odd rows > 2 are compositions of the rows 0, 1, 2, 4, 6, 8, ... (i.e. either one of the primitives A069770 or A072796, or one of the recursive compositions) at the left hand side and any Catalan bijection from the same array at the right hand side. See the scheme-functions index-for-recursive-sgtb and index-for-composed-sgtb how to compute the positions of the recursive and ordinary compositions in this table.

A. Karttunen, Gatomorphisms (With the complete source and explanation)

Four other tables giving the corresponding cycle-counts: A073201, counts of the fixed elements: A073202, the lengths of the largest cycles: A073203, the LCM's of all the cycles: A073204. The ordinary compositions are encoded using the N X N -> N bijection A054238 (which in turn uses the bit-interleaving function A000695).
The first 21 rows of this table:.
Row 0: A069770. Row 1: A072796. Row 2: A057163. Row 3: A073269, Row 4: A057163 (duplicate), Row 5: A073270, Row 6: A069767, Row 7: A001477 (identity perm.), Row 8: A069768, Row 9: A073280.
Row 10: A069770 (dupl.), Row 11: A072796 (dupl.), Row 12: A057511, Row 13: A073282, Row 14: A057512, Row 15: A073281, Row 16: A057509, Row 17: A073280 (dupl.), Row 18: A057510, Row 19: A073283, Row 20: A073284.
Other Catalan bijection-induced EIS-permutations which occur in this table. Only the first known occurrence is given. Involutions are marked with *, others paired with their inverse:.
Row 164: A057164*, Row 168: A057508*, Row 179: A072797*.
Row 41: A073286 - Row 69: A073287. Row 105: A073290 - Row 197: A073291. Row 416: A073288 - Row 696: A073289.
Row 261: A057501 - Row 521: A057502. Row 2618: A057503 - Row 5216: A057504. Row 2614: A057505 - Row 5212: A057506.
Row 10435: A073292 - Row ...: A073293. Row 17517: A057161 - Row ...: A057162.
For a more practical enumeration system of (some) Catalan automorphisms see table A089840 and its various "recursive derivations".

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

cp's OEIS Frontend

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A079226 Number of Catalan objects fixed by five-fold application of the Catalan bijections A057511/A057512 (deep rotation of general parenthesizations/plane trees).

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A079224 Number of Catalan objects fixed by three-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

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A079225 Number of Catalan objects fixed by four-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

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A130919 Signature permutation of a Catalan automorphism: DEEPEN-transform of automorphism *A057511.

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A243495 Indices in A014486 for the oriented trees that stay the same when "deep-rotated": fixed points of A057511/A057512.

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A073200 Number of simple Catalan bijections of type B.

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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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