A071248 -id:A071248 - cp's OEIS frontend

A127553 a(n) = Product_{k=1..n} lcm(k,n)/gcd(k,n).

1, 2, 18, 96, 15000, 6480, 84707280, 41287680, 21427701120, 56700000000, 94121726392108800, 2483144294400, 11159820050604594969600, 24625709514114508800, 620634514500000000000, 359450793240158011392000

Offset: 1

Leroy Quet, Apr 02 2007

nonn

			a(6) = lcm(1,6)/gcd(1,6) * lcm(2,6)/gcd(2,6) * lcm(3,6)/gcd(3,6) * lcm(4,6)/gcd(4,6) * lcm(5,6)/gcd(5,6) * lcm(6,6)/gcd(6,6) = 6/1 * 6/2 * 6/3 * 12/2 * 30/1 * 6/6 = 6480.

Harvey P. Dale, Table of n, a(n) for n = 1..238

Cf. A056789.

A067911 := proc(n) mul( gcd(k,n),k=1..n) ; end: A071248 := proc(n) mul( lcm(k,n),k=1..n) ; end: A127553 := proc(n) A071248(n)/A067911(n) ; end: for n from 1 to 30 do printf("%d, ",A127553(n)) ; od ; # R. J. Mathar, Apr 03 2007
a:=n->product(ilcm(k,n)/igcd(k,n),k=1..n): seq(a(n),n=1..18); # Emeric Deutsch, Apr 13 2007

Table[Product[LCM[k,n]/GCD[k,n],{k,n}],{n,20}] (* Harvey P. Dale, Jun 10 2017 *)

a(n) = A071248(n)/A067911(n). - R. J. Mathar, Apr 03 2007

More terms from R. J. Mathar, Apr 03 2007

A056916 Product of the orders of the elements in a cyclic group with n elements.

Original entry on oeis.org

1, 2, 9, 32, 625, 648, 117649, 131072, 4782969, 12500000, 25937424601, 214990848, 23298085122481, 1771684761728, 14416259765625, 562949953421312, 48661191875666868481, 11712917736940032, 104127350297911241532841, 5120000000000000000, 7788651757984142343081

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Feb 10 2002

Amiram Eldar, Table of n, a(n) for n = 1..388

Cf. A057660.

Cf. A067911, A071248.

a:=n->mul(denom (k/n), k=1..n): seq(a(n), n=1..18); # Zerinvary Lajos, Apr 26 2008

Table[Product[n/GCD[n,i],{i,0,n-1}],{n,30}] (* Harvey P. Dale, Oct 24 2011 *)

a(n) = Product_{ d divides n } d^phi(d). - Vladeta Jovovic, Sep 10 2004

Edited by Dean Hickerson, Mar 04 2002

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

Ilya Gutkovskiy, Jul 01 2019

nonn

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

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}]

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

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.

cp's OEIS Frontend

A127553 a(n) = Product_{k=1..n} lcm(k,n)/gcd(k,n).

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A056916 Product of the orders of the elements in a cyclic group with n elements.

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

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