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.

A071248 a(n) = Product_{k=1..n} lcm(n,k).

Original entry on oeis.org

1, 4, 54, 768, 75000, 466560, 592950960, 5284823040, 1735643790720, 45360000000000, 1035338990313196800, 102980960177356800, 145077660657859734604800, 154452450072526199193600
Offset: 1

Views

Author

Amarnath Murthy, May 21 2002

Keywords

Comments

Log(a(n))/n/Log(n) is bounded since n^n < a(n) < n^(2n). It seems that lim n -> infinity Log(a(n))/n/Log(n) exists and = 1.7.... - Benoit Cloitre, Aug 13 2002

Links

Crossrefs

Product of terms in n-th row of A051173.
Cf. A067911, A056916, A055774.

Programs

  • Maple
    A071248 := proc(n) mul( lcm(k,n),k=1..n) ; end: for n from 1 to 10 do printf("%d ",A071248(n)) ; od ; # R. J. Mathar, Apr 03 2007
  • Mathematica
    Table[Product[LCM[k,n],{k,n}],{n,20}] (* Harvey P. Dale, Jun 12 2019 *)
  • PARI
    a(n)=prod(k=1,n,lcm(n,k))

Formula

a(n) = n!*Product_{ d divides n } d^phi(d). - Vladeta Jovovic, Sep 10 2004
a(n) = n!*n^n/A067911(n)=A000142(n)*A000312(n)/A067911(n). - R. J. Mathar, Apr 03 2007

Extensions

More terms from Benoit Cloitre, Aug 13 2002

A203904 Triangular array T; for n>0, row n shows the coefficients of a reduced polynomial having zeros -k/(n+1) for k=1,2,...,n.

Original entry on oeis.org

1, 1, 2, 2, 9, 9, 3, 22, 48, 32, 24, 250, 875, 1250, 625, 10, 137, 675, 1530, 1620, 648, 720, 12348, 79576, 252105, 420175, 352947, 117649, 315, 6534, 52528, 216608, 501760, 659456, 458752, 131072, 4480, 109584, 1063116, 5450004, 16365321
Offset: 1

Views

Author

Clark Kimberling, Jan 08 2012

Keywords

Comments

For n>0, the zeros of the polynomial represented by row n+1 interlace the zeros of the polynomial for row n; see the Example section.
...
T(n,1): A119619
T(n,n): A056916.

Examples

			First five rows(counting the top row as row 0):
1
1...2.................representing 1+2x
1...9...9.............representing 2+9x+9x^2
3...22..48...32
24...250...875...1250...625
Zeros corresponding to rows 1 to 4:
.................-1/2
............-2/3......-1/3
......-3/4.......-1/2.......-1/4
-4/5........-3/5......-2/5.......-1/5
Interlace property for successive rows illustrated by
  1/5 < 1/4 < 2/5 < 1/2 < 3/5 < 3/4 < 4/5.

Crossrefs

Cf. A056856, A119619, A056916, A007305/A007306 (Farey fractions).

Programs

  • Mathematica
    p[n_, x_] := Product[(n*x + k)/GCD[n, k], {k, 1, n - 1}]
Table[CoefficientList[p[n, x], x], {n, 1, 10}]
TableForm[%]  (* A203904 triangle *)
Flatten[%%]   (* A203904 sequence *)

A308944 a(n) = Product_{k=1..n} lcm(n,k) / (k * gcd(n,k)).

Original entry on oeis.org

1, 1, 3, 4, 125, 9, 16807, 1024, 59049, 15625, 2357947691, 5184, 1792160394037, 282475249, 474609375, 17179869184, 2862423051509815793, 3486784401, 5480386857784802185939, 250000000000, 10382917022245341, 5559917313492231481, 39471584120695485887249589623
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 01 2019

Keywords

Crossrefs

Cf. A000010, A051190, A056916, A067911, A071248, A119619, A127553.

Programs

  • Mathematica
    Table[Product[LCM[n, k]/(k GCD[n, k]), {k, 1, n}], {n, 1, 23}]
Table[Product[d^(EulerPhi[d] - EulerPhi[n/d]), {d, Divisors[n]}], {n, 1, 23}]
  • PARI
    a(n) = prod(k=1, n, lcm(n, k)/(k*gcd(n, k))); \\ Michel Marcus, Jul 02 2019

Formula

a(n) = Product_{d|n} d^(phi(d)-phi(n/d)).
a(n) = n^n / Product_{d|n} d^(2*phi(n/d)).
a(n) = n^(-n) * Product_{d|n} d^(2*phi(d)).
a(n) = n^n / Product_{k=1..n} gcd(n,k)^2.
a(n) = n^(-n) * Product_{k=1..n} lcm(n,k)^2/k^2.
a(n) = A127553(n)/n!.
a(n) = A056916(n)/A067911(n).
a(p) = p^(p-2), where p is a prime.
Showing 1-3 of 3 results.

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