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

1,2

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]."

Links

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

Example

(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.

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 *)

Crossrefs

Cf. A095209, A095211.

Sequence in context: A275801 A158259 A362050 * A156469 A001545 A208954

Adjacent sequences: A095207 A095208 A095209 * A095211 A095212 A095213

Keyword

nonn

Author

Amarnath Murthy, Jun 08 2004

Extensions

More terms from Ryan Propper, Oct 10 2005

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

Status

approved