A000793 -id:A000793 - cp's OEIS frontend

A128305 a(n) is the smallest m such that g(m) is divisible by prime(n), where g is Landau's function A000793.

2, 3, 8, 14, 27, 32, 57, 62, 93, 118, 128, 178, 213, 215, 297, 346, 399, 429, 519, 510, 586, 687, 780, 920, 946, 1033, 1106, 1128, 1209, 1192, 1614, 1618, 1788, 1790, 1989, 1987, 2269, 2497, 2271, 2883, 2984, 2986, 3336, 3229, 3579, 3704, 4142, 4367, 4371

Offset: 1

Anthony C Robin, May 04 2007

nonn

			g(k) for k < 14 is not divisible by prime(4) = 7; g(14) = 84 = 7*12. Hence a(4) = 14.
g(k) for k < 32 is not divisible by prime(6) = 13; g(32) = 5460 = 13*420. Hence a(6) = 32.

Alois P. Heinz and Giovanni Resta, Table of n, a(n) for n = 1..750 (first 70 terms from Alois P. Heinz)
Jean-Pierre Massias, Jean-Louis Nicolas, 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).

Cf. A000793.

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}]]]];
g[n_] := g[n] = b[n, If[n < 8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]];
a[n_] := For[p = Prime[n]; m = 2, True, m++, If[Divisible[g[m], p], Print[n, " ", m]; Return[m]]];
Array[a, 100] (* Jean-François Alcover, Feb 19 2020, after Alois P. Heinz in A000793 *)

Edited, a(6) inserted and a(12) to a(23) added by Klaus Brockhaus, May 07 2007

a(24)-a(70) from Alois P. Heinz, Feb 16 2013

A225627 a(n) = lcm(A000793(n),p1,p2,...,pk) for such a partition {p1+p2+...+pk} of n that maximizes this value among all partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 30, 84, 120, 180, 210, 420, 660, 780, 1260, 4620, 5460, 5460, 5460, 9240, 13860, 13860, 16380, 32760, 120120, 180180, 180180, 235620, 180180, 471240, 1021020, 1021020, 1141140, 1141140, 2282280, 2282280, 4476780, 4476780, 6846840, 6846840

Offset: 0

Antti Karttunen, May 13 2013

nonn

Row 2 of A225630.

This could be called a "twice-iterated Landau's function."

Index entries for sequences related to lcm's

Cf. A000793, A225630, A225628, A225629, A225636.

(define (A225627 n) (let ((maxlcm (list 0))) (fold_over_partitions_of n (A000793 n) 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))))))))

a(n) = A225636(n)*A000793(n).

A213952 Consider the partitions of n in reverse lexicographic ordering (A080577), a(n) is the position of the partition of n which has the maximum LCM. See A000793.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 5, 8, 15, 13, 33, 49, 35, 49, 73, 107, 143, 211, 293, 398, 505, 510, 685, 710, 948, 740, 994, 2033, 1735, 2266, 1780, 2333, 4653, 5923, 7311, 9213, 7683, 9719, 17878, 14703, 19072, 22814, 28266, 34878, 42876, 52390

Offset: 1

Robert G. Wilson v, Jul 04 2012

nonn

As n grows, a(n)/P(n) -> ~1/3, where P(n) is A000041(n).

			a(5) = 3 because of the seven partitions of 5, {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}; the LCMs of each are: {5, 4, 6, 3, 2, 2, 1}. The third one is the maximum.

Robert G. Wilson v, Table of n, a(n) for n = 1..84

Cf. A000793, A080577, A000041.

f[n_] := Block[{lst = Apply[LCM, IntegerPartitions@ n, 1]}, Flatten[ Position[ lst, Max@ lst, 1, 1], 1][[1]]]; Array[f, 47]

A006644 Indices of records in Landau's function A000793.

Original entry on oeis.org

0, 2, 3, 4, 5, 7, 8, 9, 10, 12, 14, 15, 16, 17, 19, 23, 25, 27, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42, 43, 47, 49, 53, 57, 58, 59, 60, 62, 64, 66, 68, 70, 72, 76, 77, 78, 79, 83, 85, 89, 93, 95, 97, 101, 102, 106, 108, 112, 118, 120, 126, 128, 130, 131, 132

Offset: 1

N. J. A. Sloane

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3318 from Alois P. Heinz)
J.-L. Nicolas, Ordre maximum d'un élément du groupe de permutations et highly composite numbers, Bull. Math. Soc. France, 97 (1969), 129-191.

Cf. A000793, A002809.

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}]]]]; order[n_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]]; Join[{0}, Position[Differences[Array[order, 133, 0]], ?Positive] // Flatten ] (* _Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)

A129759 For the Landau function L(n), A000793, this sequence gives the largest prime which is a factor of L(n).

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 3, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, 11, 7, 11, 11, 13, 13, 11, 11, 11, 11, 13, 13, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 19, 19, 17, 17, 17, 17, 19, 19, 17, 17, 19, 19, 19, 19, 19, 19, 17, 19

Offset: 1

Anthony C Robin, May 15 2007

This function is not monotone increasing, for example a(33) = 13 while a(34) = 11.

Nicolas showed that a(n) ~ sqrt(n log n) and Grantham showed that a(n) <= 1.328 sqrt(n log n) for n > 4. Massias, Nicolas, & Robin conjecture that a(n) <= 1.265... sqrt(n log n) in this range with equality at n = 215. - Charles R Greathouse IV, Jun 02 2014

			L(29) = 2520, whose largest prime factor is 7. So a(29) = 7.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Jon Grantham, The largest prime divisor of the maximal order of an element of S_n, Math. Comp. 64:209 (1995), pp. 407-410.
J. P. Massias, J. L. Nicolas and G. Robin, Effective bounds for the maximal order of an element in the symmetric group, Math. Comp. 53:188 (1989), pp. 665-678. [alternate link]
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.
Eric Weisstein's World of Mathematics, Landau's Function

Cf. A006530, A000793, A128305.

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}]]]];
g[n_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]];
a[n_] := FactorInteger[g[n]][[-1, 1]];
Array[a, 100] (* Jean-François Alcover, Feb 19 2020, after Alois P. Heinz in A000793 *)

a(n) = A006530(A000793(n)). - R. J. Mathar, May 17 2007

More terms from Klaus Brockhaus and R. J. Mathar, May 16 2007

Corrected a(66) by Alois P. Heinz, Feb 16 2013

A226143 a(n) is the smallest k > 0 such that A000793(n)^k >= n!.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 4, 5, 5, 6, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 8, 9, 9, 9, 9, 9, 10, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 12, 13, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 15, 16, 16

Offset: 1

W. Edwin Clark, May 27 2013

nonn

This is a lower bound for A226142(n), the least positive integer k such that S_n is a product of k cyclic groups.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000793, A226142.

A000793:=
[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]:
a:=proc(n)
global A000793;
local k;
for k from 1 do
if A000793[n]^k >= n! then return k; fi;
od;
end;

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_] := Module[{m}, If[n == 1, 1, m = b[n, If[n < 8, 3, PrimePi[Ceiling[ 1.328 Sqrt[n Log[n] // Floor]]]]]; Log[m, n!] // Ceiling]];
Array[a, 100] (* Jean-François Alcover, Nov 12 2020, after Alois P. Heinz in A000793 *)

a(n) = ceiling(log_m(n!)) where m = A000793(n).

A383459 Minimum number of cycles in any permutation in S_n of the highest order (A000793(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 4, 3, 3, 3, 4, 3, 4, 5, 5, 4, 4, 5, 4

Offset: 1

Anand Jain, Mar 22 2025

Landau's function g(n) = A000793(n) gives the maximum order of any permutation on n elements.

The number of permutations of order g(n) is A074059, and the number of different cycle types of permutations of order g(n) is A074064. a(n) is the maximum number of cycles in any permutation of order g(n), and A383459(n) is the minimum number of cycles in any permutation of order g(n).

			There are two different cycle types of permutations in S_6 of the maximum order g(6) = 6, for example (123456) and (12)(345)(6). The minimum number of cycles is a(6) = 1 and maximum number is A383458(6) = 3.

Cf. A000793, A074059, A074064, A383458.

A103635 Positions of running maxima of log(g(n))/sqrt(n*log(n)), where g(n) is Landau's function A000793.

Original entry on oeis.org

2, 3, 5, 7, 9, 10, 12, 17, 19, 30, 36, 40, 43, 47, 49, 53, 60, 64, 66, 70, 83, 85, 89, 108, 112, 141, 172, 209, 250, 258, 293, 301, 321, 340, 348, 360, 368, 401, 413, 421, 480, 533, 541, 608, 626, 679, 697, 752, 770, 831, 849, 914, 932, 1021, 1118, 1160, 1219

Offset: 2

Eric W. Weisstein, Feb 11 2005

Massias proved that the function log(g(n))/sqrt(n*log(n)) reaches its maximum at n = 1319766. Therefore this sequence is finite, with a(378) = 1319766 being the last term. - Amiram Eldar, Aug 23 2019

			From _Jon E. Schoenfield_, Aug 16 2015: (Start)
Terms are the values of n at which record high values of the ratio log(g(n))/sqrt(n*log(n)) occur (where g(n) = A000793(n)):
        n  g(n)  log(g(n))/sqrt(n*log(n))
       ==  ====  ========================
        1     1        (undefined)
a(1) =  2     2          0.588705 <--- record high
a(2) =  3     3          0.605148 <--- record high
        4     4          0.588705
a(3) =  5     6          0.631623 <--- record high
        6     6          0.546467
a(4) =  7    12          0.673286 <--- record high
        8    15          0.663955
a(5) =  9    20          0.673666 <--- record high
a(6) = 10    30          0.708800 <--- record high
(End)

Amiram Eldar, Table of n, a(n) for n = 2..378 (calculated using the MAPLE code by Deléglise et al.; terms 2..123 from Alois P. Heinz)
Marc Deléglise, Jean-Louis Nicolas, and Paul Zimmermann, Landau's function for one million billions, Journal de Théorie des Nombres de Bordeaux, Vol. 20, No. 3 (2008), pp. 625-671.
Marc Deléglise, Jean-Louis Nicolas, and Paul Zimmermann, Computation of the Landau function g(n) (MAPLE code).
Jean-Pierre Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique, Annales de la Faculté des sciences de Toulouse: Mathématiques, Vol. 6, No. 3-4 (1984), pp. 269-281.
Eric Weisstein's World of Mathematics, Landau's Function

Cf. A000793.

g[n_] := Max@Apply[LCM, IntegerPartitions@n, 1]; f[n_] := Log[g[n]]/Sqrt[n * Log[n]]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 2, 100}]; s (* Amiram Eldar, Aug 23 2019 after Robert G. Wilson v at A000793 *)

More terms from R. J. Mathar, Feb 14 2008

More terms from Alois P. Heinz, Feb 18 2013

A174234 A variant of Landau's function (A000793) with a restriction on the length of cycles. a(n) is the largest value of lcm(p_1, ..., p_k), with p_1 + ... + p_k <= n, such that there exist integer offsets f_1, ..., f_k with 0 <= f_i < p_i, for which f_i and f_j are different modulo gcd(p_i, p_j).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 12, 12, 13, 24, 24, 30, 30, 40, 40, 42, 42, 60, 60, 70, 70, 84, 84, 90, 90, 120, 120, 126, 126, 168, 168, 180, 180, 240, 240, 240, 240, 336, 336, 336, 336, 420, 420, 420, 420, 560, 560, 560, 560, 720, 720, 720, 720, 880, 880, 880

Offset: 1

Alexander Okhotin (alexander.okhotin(AT)utu.fi), Mar 13 2010

nonn

a(n) is the maximal period of any set of nonintersecting congruences with moduli summing to at most n. - Charlie Neder, May 09 2019

			a(10)=12 is given by k=2, p_1=4, p_2=6, f_1=0 and f_2=1, with 0 != 1 mod(gcd(4, 6)).

Charlie Neder, Table of n, a(n) for n = 1..150
A. Okhotin, "A study of unambiguous finite automata over a one-letter alphabet"

Cf. Landau's function (A000793).

Asymptotic: log a(n) ~ (n log(n)^2) ^ 1/3.

a(51) onwards and minor edits from Charlie Neder, May 09 2019

A211391 The number of divisors d of n! such that d < A000793(n) (Landau's function g(n)) and the symmetric group S_n contains no elements of order d.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 2, 2, 6, 4, 15, 15, 24, 29, 33, 63, 55, 126, 117, 110, 103, 225, 212, 288, 282, 319, 428, 504, 774, 859, 943, 924, 1336, 1307, 1681, 1869, 2097, 2067, 2866, 3342, 3487, 5612, 5567, 5513, 5549, 9287, 9220, 11594, 11524, 11481, 11403, 18690

Offset: 1

Alexander Gruber, Feb 07 2013

nonn

This sequence gives the number of divisors d of |S_n| such that d < Lambda(n) (where Lambda(n) = the largest order of an element in S_n) for which S_n contains no element of order d. These divisors constitute a set of 'missing' element orders of S_n.

For computational purposes, the smallest divisor d0(n) of n! = |S_n| for which S_n has no element of order d0(n) is the smallest divisor of n! which is not the least common multiple of an integer partition of n. Thus d0(n) is given by the smallest prime power >= n+1 that is not prime (with the exception of n = 3 and 4, for which d0(n) = 6).

			For n = 7, we refer to the following table:
Symmetric Group on 7 letters.
  # of elements of order  1 ->    1
  # of elements of order  2 ->  231
  # of elements of order  3 ->  350
  # of elements of order  4 ->  840
  # of elements of order  5 ->  504
  # of elements of order  6 -> 1470
  # of elements of order  7 ->  720
  # of elements of order  8 ->    0
  # of elements of order  9 ->    0
  # of elements of order 10 ->  504
  # of elements of order 12 ->  420
  (All other divisors of 7! -> 0.)
So there are two missing element orders in S_7, whence a(7) = 2.

d0(n) is equal to A167184(n) for n >= 5.

Cf. A000793 (Landau's function g(n)), A057731, A211392.

for n in [1..25] do
D := Set(Divisors(Factorial(n)));
O := { LCM(s) : s in Partitions(n) };
L := Max(O);
N := D diff O;
#{ n : n in N | n lt L };
end for;

More terms from Alois P. Heinz, Feb 11 2013

cp's OEIS Frontend

A128305 a(n) is the smallest m such that g(m) is divisible by prime(n), where g is Landau's function A000793.

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A225627 a(n) = lcm(A000793(n),p1,p2,...,pk) for such a partition {p1+p2+...+pk} of n that maximizes this value among all partitions of n.

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A213952 Consider the partitions of n in reverse lexicographic ordering (A080577), a(n) is the position of the partition of n which has the maximum LCM. See A000793.

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A006644 Indices of records in Landau's function A000793.

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A129759 For the Landau function L(n), A000793, this sequence gives the largest prime which is a factor of L(n).

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A226143 a(n) is the smallest k > 0 such that A000793(n)^k >= n!.

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Maple

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A383459 Minimum number of cycles in any permutation in S_n of the highest order (A000793(n)).

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A103635 Positions of running maxima of log(g(n))/sqrt(n*log(n)), where g(n) is Landau's function A000793.

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