A000793 -id:A000793 - cp's OEIS frontend

A159685 Maximal product of distinct primes whose sum is <= n.

1, 2, 3, 3, 6, 6, 10, 15, 15, 30, 30, 42, 42, 70, 105, 105, 210, 210, 210, 210, 330, 330, 462, 462, 770, 1155, 1155, 2310, 2310, 2730, 2730, 2730, 2730, 4290, 4290, 6006, 6006, 10010, 15015, 15015, 30030, 30030, 30030, 30030, 39270, 39270, 46410, 46410

Offset: 1

Wouter Meeussen, Apr 19 2009, May 02 2009

nonn

Equivalently, largest value of the LCM of the partitions of n into primes.
Equivalently, maximal number of times a permutation of length n, with prime cycle lengths, can operate on itself before returning to the initial permutation.
If the requirement that primes are distinct is dropped, this becomes A000792. - Charles R Greathouse IV, Jul 10 2012

			A permutation of length 10 can have prime cycle lengths of 2+3+5; so when repeatedly applied to itself, can produce at most 2*3*5 different permutations.
The products of distinct primes whose sum is <= 10 are 1 (the empty product), 2, 3, 5, 7, 2*3=6, 2*5=10, 2*7=14, 3*5=15, 3*7=21, and 2*3*5=30. The maximum is 30, so a(10) = 30. - _Jonathan Sondow_, Jul 06 2012

Alois P. Heinz, Table of n, a(n) for n = 1..10000
M. Deléglise and J.-L. Nicolas, Maximal product of primes whose sum is bounded, arXiv 1207.0603 [math.NT] (2012).
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, Jean-Louis Nicolas, The Landau function and the Riemann hypothesis, Univ. Lyon (France, 2019).

Cf. A077011, A000793, A034891.

with(numtheory):
b:= proc(n,i) option remember; local p; p:= ithprime(max(i,1));
      `if`(n=0, 1, `if`(i<1, 0,
       max(b(n, i-1), `if`(p>n, 0, b(n-p, i-1)*p))))
    end:
a:= proc(n) option remember;
     `if`(n=0, 1, max(b(n, pi(n)), a(n-1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 04 2012

temp=Series[Times @@ (1/(1-q[ # ] x^#)& /@ Prepend[Prime /@ Range[24],1]),{x,0,Prime[24]}]; Table[Max[List @@ Expand[Coefficient[temp,x^n]]/. q[a_]^_ ->q[a] /.q->Identity],{n,64}]
(* Second program: *)
b[n_, i_] := b[n, i] = Module[{p = Prime[Max[i, 1]]}, If[n == 0, 1, If[i < 1, 0, Max[b[n, i-1], If[p > n, 0, b[n-p, i-1]*p]]]]]; a[n_] := a[n] = If[n == 0, 1, Max[b[n, PrimePi[n]], a[n-1]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 05 2013, translated from Alois P. Heinz's Maple program *)

a(n) <= A002809(n) and A008475(a(n)) <= n (see (1.2) and (1.4) in Deléglise-Nicolas 2012). - Jonathan Sondow, Jul 04 2012.

A225646 a(n) = lcm(n,p1,p2,...,pk) for such a partition of n which maximizes this value among all partitions {p1+p2+...pk} of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 30, 84, 120, 180, 210, 330, 420, 780, 630, 840, 1680, 3570, 1386, 7980, 1980, 4620, 6930, 19320, 9240, 23100, 30030, 41580, 16380, 73080, 10920, 143220, 110880, 120120, 157080, 120120, 180180, 512820, 311220, 240240, 360360, 1231230, 180180

Offset: 0

Antti Karttunen, May 15 2013

nonn

Second row of table A225640.

a(0)=1 by convention.

Index entries for sequences related to lcm's

Cf. A225640, A225642, A225652, A000793, A225655.

(define (A225646 n) (let ((maxlcm (list 1))) (fold_over_partitions_of n 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))))))))

A290932 Sum of the LCM of cycle lengths over all endofunctions on [n].

Original entry on oeis.org

1, 1, 5, 40, 431, 5886, 96817, 1862890, 41043375, 1018584610, 28108489541, 853617865134, 28287119604955, 1015630741097350, 39273014068691145, 1627118268024495586, 71904849762914854703, 3375959341815207350850, 167810405947367539063885, 8803814897608815310714270

Offset: 0

Alois P. Heinz, Aug 13 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A000793, A060014, A208231, A208248, A222029.

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, ilcm(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> add(b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

T[n_, k_] := T[n, k] = If[n == 0, k, Sum[(j - 1)! * T[n - j, LCM[k, j]]*Binomial[n - 1, j - 1], {j, n}]]; {1}~Join~Table[Sum[T[j, 1]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}], {n, 19}] (* Michael De Vlieger, Aug 17 2017 *)

a(n) = Sum_{k=1..A000793(n)} k * A222029(n,k).

A113309 a(n) = the number of finite sequences of positive integers {b(k)} where (product b(k)) * (sum b(k)) = n. Different orderings of the same sequence {b(k)} are not counted separately.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 5, 1, 4, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 4, 3, 2, 1, 9, 2, 3, 2, 4, 1, 6, 2, 7, 2, 2, 1, 8, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 11, 1, 2, 3, 4, 2, 5, 1, 9, 4, 2, 1, 10, 2, 2, 2, 7, 1, 9, 2, 4, 2, 2, 2, 13, 1, 3, 4, 7, 1, 5, 1, 7

Offset: 1

Leroy Quet, Oct 25 2005

nonn

Sequence's terms calculated by "Max".

First occurrence: 1, 4, 12, 16, 24, 54, 36, 60, 48, 84, 72, 108, 96, ..., . - Robert G. Wilson v, May 03 2006

			6 = (1*1*1*1*1*1) * (1+1+1+1+1+1) = (1*2) * (1+2). So a(6) = 2.

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 4096 terms from Antti Karttunen)

Cf. A057567, A113308.

t = Table[1, {104}]; Do[k = 1; lmt = PartitionsP[n]; p = IntegerPartitions[n]; While[k < lmt, a = Plus @@ p[[k]]*Times @@ p[[k]]; If[a < 105, t[[a]]++ ]; k++ ], {n, 52}]; t (* Robert G. Wilson v, May 03 2006 *)

(define (A113309 n) (let ((z (list 0))) (let loop ((k n)) (cond ((zero? k) (car z)) ((not (zero? (modulo n k))) (loop (- k 1))) (else (begin (fold_over_partitions_with_uplim_cut k 1 * (lambda (partprod) (if (= n (* k partprod)) (set-car! z (+ 1 (car z))))) (/ n k)) (loop (- k 1))))))))
(define (fold_over_partitions_with_uplim_cut m initval addpartfun colfun uplim) (let recurse ((m m) (b m) (n 0) (partition initval)) (cond ((zero? m) (colfun partition)) ((> partition uplim) #f) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (addpartfun i partition)) (if (< i (min b m)) (loop (+ 1 i)))))))) ;; This function is a modification of fold_over_partitions_of given in A000793.
;; Antti Karttunen, Nov 03 2017

a(n) = 1 iff n = 1 or n is a prime. a(n) = 2 if n is a semiprime. - Robert G. Wilson v, May 03 2006

a(n) = Sum_{d|n} {number of partitions of d where product of parts = n/d}. - Antti Karttunen, Nov 03 2017

More terms from Robert G. Wilson v, May 03 2006

A129651 a(n) is the smallest position k at which b_n(i)=k, where b_n(m) is the largest order of a permutation of m elements with exactly n cycles.

Original entry on oeis.org

1, 6, 37, 126, 287, 540, 895

Offset: 1

Nickolas Reynolds (nickels(AT)gmail.com), Apr 25 2007, Apr 26 2007, Apr 27 2007

			a(2)=6 because b_2(6)=5 and b_2(i)<b_2(i+1) for all i>=6. (That is, the largest order of a permutation of i elements with exactly 2 cycles is monotonic increasing starting at i=6.)

Cf. A000793, A129647, A129648, A129649, A129650.

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).

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).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 10, 12, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 21, 30

Offset: 0

Alois P. Heinz, Apr 01 2015

			Triangle T(n,k) begins:
  1;
  1;
  1, 2;
  1, 2, 3;
  1, 2, 3, 4;
  1, 2, 3, 4, 5, 6;
  1, 2, 3, 4, 5, 6;
  1, 2, 3, 4, 5, 6, 7, 10, 12;
  1, 2, 3, 4, 5, 6, 7,  8, 10, 12, 15;
  1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 12, 14, 15, 20;
  1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 12, 14, 15, 20, 21, 30;

Alois P. Heinz, Rows n = 0..60, flattened

Row sums give A060179.
Row lengths give A009490.
Last elements of rows give A000793.
Main diagonal gives A000027.
Cf. A181844, A256067, A256554.

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((h->`if`(h=0, [][], i))(coeff(p, x, i))
     , i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);

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[Function[h, If[h == 0, Nothing, i]][
     Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jul 15 2021, after Alois P. Heinz *)

Sum_{k>=0} T(n,k)*A256554(n,k) = A181844(n).

T(n,k) = k for n>0 and 1<=k<=n.

A082342 Permutation of natural numbers induced by the Catalan bijection gma082342 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 12, 11, 13, 17, 18, 16, 14, 15, 21, 19, 20, 22, 23, 24, 26, 25, 27, 31, 32, 30, 28, 29, 35, 33, 34, 36, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 39, 40, 41, 58, 59, 56, 51, 52, 57, 53, 54, 55, 63, 60, 61, 62, 64, 65, 66, 68, 67, 69

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

This is A057163-conjugate of A073284.

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082341. a(n) = A057163(A073284(A057163(n))). Occurs in A073200. Cf. also A072797, A082337-A082339.
Differs from A082325 first time at n=39: a(39)=48, while A082325(39)=49.
Number of cycles: A057513. Number of fixed-points: A057546. Max. cycle size: A000793. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A123131 Largest order of permutations of n elements with no fixed points.

Original entry on oeis.org

2, 3, 4, 6, 6, 12, 15, 20, 30, 30, 60, 42, 84, 105, 140, 210, 210, 420, 280, 420, 420, 840, 504, 1260, 1155, 1540, 2310, 2520, 4620, 3080, 5460, 4620, 9240, 5544, 13860, 9240, 16380, 15015, 27720, 30030, 32760, 60060, 40040, 60060, 60060, 120120, 72072, 180180

Offset: 2

Antoine Verroken and Vladeta Jovovic, Sep 30 2006

nonn

			For n=22 we have a(22)=420 since 22 = 4 + 5 + 6 + 7 = 3 + 3 + 4 + 5 + 7 and lcm([4, 5, 6, 7]) = lcm([3, 3, 4, 5, 7]) = 420.
For n=26 we have a(26)=1155 since 26 = 3 + 5 + 7 + 11 and lcm([3,5,7,11]) = 1155.

Gheorghe Coserea, Table of n, a(n) for n = 2..124
Gheorghe Coserea, Partitions solutions for n = 2..124
J.-L. Nicolas, Ordre maximal d’un élément du groupe S_n des permutations et «highly composite numbers», Bull. Math. Soc. France, 97 (1969), 129-191.

Cf. A000793.

seq(N) = {
  my(a = vector(N+1,n,n));
  for (n=5, #a, forpart(p=n, a[n] = max(a[n],lcm(Vec(p))), [2, n-2]));
  a[2..#a];
};
seq(48) \\ Gheorghe Coserea, Dec 22 2017

From Gheorghe Coserea, Dec 24 2017: (Start)
A000793(n-2) <= a(n) <= A000793(n), for all n >= 4.
If A000793(n-1) < A000793(n) then a(n) = A000793(n).
(End)

A162682 If S is countable finite set, we can define n as number of elements in S. There are n^n distinct functions f(S)->S. Each function has a fixed point, or an orbit in S. This sequence is a number of distinct functions g(S)->S, with largest orbit.

Original entry on oeis.org

1, 1, 1, 2, 6, 20, 840, 420, 2688, 18144, 120960, 15966720, 7983360, 1349187840, 1037836800, 12454041600, 149448499200, 1693749657600, 579262382899200, 289631191449600, 115852476579840000, 26822744640147456000, 4750241170964889600000, 30776210403434496000

Offset: 0

Dmitriy Samsonov (dmitriy.samsonov(AT)gmail.com), Jul 10 2009

nonn

Sizes of orbits are given by A000793.

			For S={a}, n=1 and only one operation possible {a->a}. For S={a,b}, n=2 and possible operations are {a->a,b->a}, {a->a,b->b}, {a->b,b->a},{a->b,b->b}. Longest orbit generated by applying operation {a->b,b->a}: initial set (a,b), applying function gives orbit - (b,a), (a,b). All other possible functions are generating fixed points.

Alois P. Heinz, Table of n, a(n) for n = 0..130

Cf. A000793, A074115, A074859, A222029.

a(n) = A222029(n,A000793(n)). - Alois P. Heinz, Aug 14 2017

a(0), a(10)-a(23) from Alois P. Heinz, Jul 12 2017

a(21)-a(22) corrected by Alois P. Heinz, Aug 16 2017

cp's OEIS Frontend

A159685 Maximal product of distinct primes whose sum is <= n.

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A225646 a(n) = lcm(n,p1,p2,...,pk) for such a partition of n which maximizes this value among all partitions {p1+p2+...pk} of n.

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A290932 Sum of the LCM of cycle lengths over all endofunctions on [n].

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A113309 a(n) = the number of finite sequences of positive integers {b(k)} where (product b(k)) * (sum b(k)) = n. Different orderings of the same sequence {b(k)} are not counted separately.

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A129651 a(n) is the smallest position k at which b_n(i)=k, where b_n(m) is the largest order of a permutation of m elements with exactly n cycles.

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

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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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A082342 Permutation of natural numbers induced by the Catalan bijection gma082342 acting on the parenthesizations encoded by A014486/A063171.

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A123131 Largest order of permutations of n elements with no fixed points.

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