cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 41-50 of 80 results. Next

A129648 Largest order of a permutation of n elements with exactly 3 cycles. Also the largest LCM of a 3-partition of n.

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 6, 12, 15, 30, 21, 60, 35, 84, 105, 140, 84, 210, 165, 280, 315, 360, 385, 504, 495, 630, 693, 792, 819, 990, 1001, 1170, 1287, 1430, 1365, 1716, 1683, 2002, 2145, 2310, 2431, 2730, 2805, 3120, 3315, 3570, 3705, 4080, 4199, 4560, 4845, 5168
Offset: 1

Views

Author

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

Keywords

Comments

a(n) is asymptotic to (n^3)/27.

Examples

			a(9) = 15 because 9 = 5+3+1 and lcm(1,3,5) = 15 is maximal.

Links

Crossrefs

Cf. A000793, A129651.
Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), this sequence (k=3), A129649 (k=4), A129650 (k=5), A355367 (k=6), A355403 (k=7).

Programs

  • Mathematica
    Max[LCM @@@ Compositions[ #, 3]] & /@ Range[1, n]

A129649 Largest order of a permutation of n elements with exactly 4 cycles. Also the largest LCM of a 4-partition of n.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 6, 6, 12, 15, 30, 30, 60, 60, 84, 105, 210, 140, 420, 210, 330, 420, 840, 420, 1260, 1155, 1540, 1365, 2520, 1320, 3080, 3465, 3960, 4095, 5544, 5005, 6930, 6435, 8190, 9009, 10296, 8415, 12870, 11781, 13464, 15015, 18018, 17017, 20592, 21879
Offset: 1

Views

Author

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

Keywords

Comments

a(n) is asymptotic to (n^4)/256.

Examples

			a(18)=140 because 18 = 7+5+2+2 and lcm(2,2,5,7) = 140 is maximal.

Links

Crossrefs

Cf. A000793, A129651.
Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), A129648 (k=3), this sequence (k=4), A129650 (k=5), A355367 (k=6), A355403 (k=7).

Programs

  • Mathematica
    Max[LCM @@@ Compositions[ #, 4]] & /@ Range[1, n] (* needs Combinatorica *)
Join[{0,0,0},Table[Max[LCM@@#&/@IntegerPartitions[n,{4}]],{n,4,50}]] (* Harvey P. Dale, Feb 25 2012 *)

A129650 Largest order of a permutation of n elements with exactly 5 cycles. Also the largest LCM of a 5-partition of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 6, 12, 15, 30, 30, 60, 60, 84, 105, 210, 210, 420, 420, 420, 420, 840, 840, 1260, 1260, 2310, 1540, 4620, 2520, 5460, 4620, 9240, 5460, 13860, 9240, 16380, 15015, 27720, 13860, 32760, 19635, 40040, 45045, 51480, 32760, 72072, 58905
Offset: 1

Views

Author

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

Keywords

Comments

a(n) is asymptotic to n^5/3125.

Examples

			a(29)=1540 because 29 = 11+7+5+4+2 and lcm(2,4,5,7,11) = 1540 is maximal.

Links

Crossrefs

Cf. A000793, A129651.
Maximal LCM of k positive integers with sum n for k = 2..7: A129647 (k=2), A129648 (k=3), A129649 (k=4), this sequence (k=5), A355367 (k=6), A355403 (k=7).

Programs

  • Mathematica
    Max[LCM @@@ Compositions[ #, 5]] & /@ Range[1, n]

A051703 Maximal value of products of partitions of n into powers of distinct primes (1 not considered a power).

Original entry on oeis.org

1, 0, 2, 3, 4, 6, 0, 12, 15, 20, 30, 28, 60, 40, 84, 105, 140, 210, 180, 420, 280, 330, 360, 840, 504, 1260, 1155, 1540, 2310, 2520, 4620, 3080, 5460, 3960, 9240, 5544, 13860, 6552, 16380, 15015, 27720, 30030, 32760, 60060, 40040, 45045, 51480, 120120
Offset: 0

Views

Author

Vladeta Jovovic

Keywords

Examples

			a(11) = 28 because max{11, 2*3^2, 2^3*3, 2^2*7} = 28.

Links

Crossrefs

Largest element of n-th row of A080743.
A000793(n)=max{A000793(n-1), a(n)}, A000793(0)=1.
Cf. A008475, A051613, A080743, A080744, A051704.

Programs

  • Maple
    b:= proc(n, i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0, 1, `if`(i<1 or n<0, 0, max(b(n, i-1),
      seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))) ))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 16 2013
  • Mathematica
    nmax = 48; Do[a[n]=0, {n, 1, nmax}]; km = PrimePi[nmax]; For[k=1, k <= km, k++, q = 1; p = Prime[k]; For[i=nmax, i >= 1, i--, q=1; While[q*p <= i, q *= p; If[i == q, m = q, If[a[i - q] != 0, m = q*a[i - q], m = 0]]; a[i] = Max[a[i], m]]]]; a[0] = 1; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Aug 02 2012, translated from Robert Gerbicz's Pari program *)
  • PARI
    {N=1000;v=vector(N,i,0);forprime(p=2,N,q=1;forstep(i=N,1,-1,
q=1;while(q*p<=i,q*=p;if(i==q,M=q,if(v[i-q],M=q*v[i-q],M=0));
v[i]=max(v[i],M))));print(0" "1);for(i=1,N,print(i" "v[i]))} \\ Robert Gerbicz, Jul 31 2012

Extensions

Corrected and extended by Robert Gerbicz, Jul 31 2012

A073203 Array of maximum cycle length sequences for the table A073200.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 2, 2, 6, 2, 2, 1, 1, 2, 2, 2, 8, 2, 3, 2, 1, 1, 2, 2, 2, 10, 2, 6, 4, 1, 1, 1, 2, 2, 2, 12, 2, 8, 8, 1, 2, 1, 1, 2, 2, 2, 14, 2, 10, 16, 1, 4, 1, 1, 1, 2, 2, 2, 16, 2, 12, 32, 1, 8, 2, 2, 1, 1
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

Each row of this table gives the longest cycle/orbit produced by the Catalan bijection (given in the corresponding row of A073200) when it acts on A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Links

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

Crossrefs

Cf. also A073201, A073202, A073204.
Few EIS-sequences which occur in this table. Only the first known occurrence(s) given:.
Rows 6 and 8: A011782, Row 7: A000012, Row 12, 14: A000793 (shifted right and prepended with 1), Row 261: A057543, Row 2614: A057545, Rows 2618, 17517: A057544.

A082325 Permutation of natural numbers: A057163-conjugate of A057511.

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, 49, 48, 50, 44, 47, 42, 37, 38, 43, 40, 39, 41, 58, 59, 56, 51, 52, 57, 53, 54, 55, 63, 60, 61, 62, 64, 65, 66, 68, 67, 69
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082326. a(n) = A069787(A082326(A069787(n))). a(n) = A082327(A082853(n))+A082852(n). Occurs in A073200 as row 1792. Cf. also A082337-A082338.
Differs from A082342 first time at n=39: a(39)=49, while A082342(39)=48.
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).

Formula

a(n) = A057163(A057511(A057163(n)))

A082326 Permutation of natural numbers: A057163-conjugate of A057512.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082325. a(n) = A069787(A082325(A069787(n))). a(n) = A082328(A082853(n))+A082852(n). Occurs in A073200 as row 1794. Cf. also A082337-A082338.
Differs from A082341 first time at n=39: a(39)=49, while A082341(39)=48.
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).

Formula

a(n) = A057163(A057512(A057163(n)))

A225650 The greatest common divisor of Landau g(n) and n.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 1, 1, 1, 10, 1, 12, 1, 14, 15, 4, 1, 6, 1, 20, 21, 2, 1, 24, 5, 2, 1, 14, 1, 30, 1, 4, 3, 2, 35, 36, 1, 2, 39, 40, 1, 42, 1, 44, 15, 2, 1, 24, 7, 10, 3, 52, 1, 18, 55, 56, 3, 2, 1, 60, 1, 2, 21, 8, 65, 66, 1, 4, 3, 70, 1, 72, 1, 2, 15, 76, 77, 78, 1
Offset: 0

Views

Author

Antti Karttunen, May 11 2013

Keywords

Links

Crossrefs

A225648 gives the position of ones, and likewise A225651 gives the positions of fixed points, that is, a(A225651(n)) = A225651(n) for all n.
Cf. A225640-A225643, A225654.

Programs

  • Mathematica
    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_] := GCD[n, g[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 02 2016, after Alois P. Heinz *)
  • Scheme
    (define (A225650 n) (gcd (A000793 n) n))
;; Scheme-code for A000793 can be found in the Program section of that entry.

Formula

a(n) = gcd(n, A000793(n)).

A225655 a(n) = largest LCM of partitions of n divisible by n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 11, 60, 13, 84, 105, 16, 17, 180, 19, 420, 420, 330, 23, 840, 25, 780, 27, 1540, 29, 4620, 31, 32, 4620, 3570, 9240, 13860, 37, 7980, 16380, 27720, 41, 32760, 43, 60060, 45045, 19320, 47, 55440, 49, 23100, 157080, 180180, 53
Offset: 1

Views

Author

Antti Karttunen, May 19 2013

Keywords

Comments

a(n) = lcm(p1,p2,...,pk) for that partition of n for which the LCM is a multiple of n, and which maximizes this value among all such partitions [p1,p2,...,pk] of n.

Links

Crossrefs

For all n, a(A225651(n)) = A000793(A225651(n)).
Cf. A225646, A225656, A225652.
A225657 lists the values of n for which a(n) = n.

Programs

  • Maple
    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:
a:= n-> max(select(x-> irem(x, n)=0, b(n$2))[]):
seq(a(n), n=1..50);  # Alois P. Heinz, May 26 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Union @ Flatten @ Table[ Map[ Function[{x}, LCM[x, If[j==0, 1, i]]], b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := Max[Select[b[n, n], Mod[#, n]==0&]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

A082341 Permutation of natural numbers induced by the Catalan bijection gma082341 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, 20, 21, 19, 22, 23, 24, 26, 25, 27, 31, 32, 30, 28, 29, 34, 35, 33, 36, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 39, 40, 41, 54, 55, 57, 58, 59, 53, 56, 51, 52, 61, 62, 63, 60, 64, 65, 66, 68, 67, 69
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This is A057163-conjugate of A073285.

Links

Crossrefs

Inverse of A082342. a(n) = A057163(A073285(A057163(n))). Occurs in A073200 as row 1800. Cf. also A072797, A082337-A082339.
Differs from A082326 first time at n=39: a(39)=48, while A082326(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).
Previous Showing 41-50 of 80 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.