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A095210
a(n) = least multiple of n such that the geometric mean of a(1), ..., a(n) is an integer.
3
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]."
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
Sequence in context: A275801 A158259 A362050 * A156469 A001545 A208954
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