A056916 -id:A056916

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A071248

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

T. D. Noe, Table of n, a(n) for n=1..100

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

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 *)

PROG

(PARI) a(n)=prod(k=1, n, lcm(n, k))

CROSSREFS

Product of terms in n-th row of A051173.

Cf. A067911, A056916, A055774.

KEYWORD

nonn

AUTHOR

Amarnath Murthy, May 21 2002

EXTENSIONS

More terms from Benoit Cloitre, Aug 13 2002

STATUS

approved

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.

+10
2

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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.

LINKS

Table of n, a(n) for n=1..41.

EXAMPLE

First five rows(counting the top row as row 0):

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.

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 *)

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jan 08 2012

STATUS

approved

A308944

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..23.

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.

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}]

PROG

(PARI) a(n) = prod(k=1, n, lcm(n, k)/(k*gcd(n, k))); \\ Michel Marcus, Jul 02 2019

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Jul 01 2019

STATUS

approved

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