A095210 a(n) = least multiple of n such that the geometric mean of a(1), ..., a(n) is an integer.

1, 4, 54, 96, 37500, 60, 49412580, 107520, 16533720, 2520, 718985409939720, 27720, 8395697954737253160, 360360, 360360, 23616552960, 596208601546720632677647440, 12252240, 24240072441867520569208380462960, 232792560, 232792560, 232792560, 4860817599682675053132316060135142981520

Offset: 1

Author

Amarnath Murthy, Jun 08 2004

Keywords

• nonn

Comments

a(11), if it exists, is greater than 10^12. - Ryan Propper, Oct 10 2005

Comments from Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 25 2005: "Sequence is infinite. For a prime p, a(p) has p^p as a factor. Factoring the a(n) gives the pattern for the exponents:

[2, 1]

[2, 2]

[2, 1; 3, 3]

[2, 5; 3, 1]

[2, 2; 3, 1; 5, 5]

[2, 2; 3, 1; 5, 1]

[2, 2; 3, 1; 5, 1; 7, 7]

[2, 10; 3, 1; 5, 1; 7, 1]

[2, 3; 3, 10; 5, 1; 7, 1]

[2, 3; 3, 2; 5, 1; 7, 1]

[2, 3; 3, 2; 5, 1; 7, 1; 11, 11]

[2, 3; 3, 2; 5, 1; 7, 1; 11, 1]

[2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 13, 13]

[2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]

[2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]

[2, 19; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]

[2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 17]

[2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]

[2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 19]

[2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1]

[2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1]

[2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1]

[2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1; 23, 23]."

Examples

			(1*4*54*96)^(1/4) = (20736)^(1/4) = 12.
a(5) = 37500 = 2^2 * 3 * 5^5.
a(11) = 718985409939720 = 2^3 * 3^2 * 5 * 7 * 11^11.
		

Crossrefs

Cf. A095209, A095211.

Programs

Mathematica

p = 1; Do[k = 1; While[ !IntegerQ[(p*k*n)^(1/n)], k++ ]; Print[k*n]; p *= (k*n), {n, 1, 10}] (* Ryan Propper, Oct 10 2005 *)

Extensions

More terms from Ryan Propper, Oct 10 2005

a(11) onwards from Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 25 2005