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.

Showing 1-3 of 3 results.

A093880 a(n) = lcm(1, 2, ..., 2n) / lcm(1, 2, ..., n).

Original entry on oeis.org

2, 6, 10, 70, 42, 462, 858, 858, 4862, 92378, 8398, 193154, 74290, 222870, 6463230, 200360130, 11785890, 11785890, 22951470, 22951470, 941010270, 40463441610, 1759280070, 82686163290, 115760628606, 115760628606, 2045104438706
Offset: 1

Views

Author

Amarnath Murthy, Apr 22 2004

Keywords

Comments

Also, lcm(n+1, n+2, ..., 2n-1, 2n) / lcm(1, 2, ..., n-1, n).

Examples

			The LCM of {1,2,3,4,5,6} is 60 and the LCM of {1,2,3} is 6, so a(3) = 60/6 = 10.

Links

Crossrefs

Cf. A080397.

Programs

  • Maple
    a:=n->lcm(seq(j,j=n+1..2*n))/lcm(seq(j,j=1..n)): seq(a(n),n=1..32); # Emeric Deutsch, Feb 02 2006
  • Mathematica
    f[n_] := LCM @@ Table[i, {i, 2n}]/LCM @@ Table[i, {i, n}]; Table[ f[n], {n, 27}] (* Robert G. Wilson v, Jan 22 2005 *)

Formula

The prime number theorem implies that a(n) = e^(n(1+o(1))) as n -> infinity. In other words, log(a(n))/n -> 1 as n -> infinity. - Jonathan Sondow, Jan 17 2005
a(n) = A003418(2n)/A003418(n) = A099996(n)/A003418(n).

Extensions

More terms from Emeric Deutsch, Feb 02 2006
Entry revised by N. J. A. Sloane, Jan 24 2007

A163641 The radical of the swinging factorial A056040.

Original entry on oeis.org

1, 1, 2, 6, 6, 30, 10, 70, 70, 210, 42, 462, 462, 6006, 858, 4290, 4290, 72930, 24310, 461890, 92378, 1939938, 176358, 4056234, 1352078, 6760390, 520030, 1560090, 222870, 6463230, 6463230, 200360130
Offset: 0

Views

Author

Peter Luschny, Aug 02 2009

Keywords

Comments

The radical of n$ is the product of the prime numbers dividing n$. It is the largest squarefree divisor of n$, and so also described as the squarefree kernel of n$.

Examples

			11$ = 2772 = 2^2*3^2*7*11. Therefore a(11) = 2*3*7*11 = 462.

Links

Crossrefs

Bisections give: A080397 (even part), A163640 (odd part).
Cf. A056040.

Programs

  • Maple
    a := proc(n) local p; mul(p,p=numtheory[factorset](n!/iquo(n,2)!^2)) end:
  • Mathematica
    sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[0] = 1; a[n_] := Times @@ FactorInteger[sf[n]][[All, 1]]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jul 26 2013 *)

Formula

a(n) = rad(n$).

A163640 The radical of the swinging factorial A056040 for odd indices.

Original entry on oeis.org

1, 6, 30, 70, 210, 462, 6006, 4290, 72930, 461890, 1939938, 4056234, 6760390, 1560090, 6463230, 200360130, 2203961430, 907513530, 33578000610, 22974421470, 941951280270, 5786272150230, 526024740930, 1074920122770, 7524440859390, 25583098921926, 104300326374006, 1912172650190110
Offset: 0

Views

Author

Peter Luschny, Aug 02 2009

Keywords

Comments

Let $ denote the swinging factorial. a(n) is the radical of (2*n+1)$ which is the product of the prime numbers dividing (2*n+1)$. It is the largest squarefree divisor of (2*n+1)$, and so also described as the squarefree kernel of (2*n+1)$.

Examples

			(2*5+1)$ = 2772 = 2^2*3^2*7*11. Therefore a(5) = 2*3*7*11 = 462.

Links

Crossrefs

A056040(n) = n$, A163641(n) = rad(n$), A080397(n) = rad((2n)$).

Programs

  • Maple
    a := proc(n) local p; mul(p,p=numtheory[factorset]((2*n+1)!/iquo(2*n+1,2)!^2)) end:
  • Mathematica
    sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Times @@ FactorInteger[sf[2*n + 1]][[All, 1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 30 2013 *)

Extensions

More terms from Michel Marcus, Aug 22 2025
Showing 1-3 of 3 results.

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

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