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%I #8 Nov 18 2018 00:05:36
%S 1,4,54,96,37500,60,49412580,107520,16533720,2520,718985409939720,
%T 27720,8395697954737253160,360360,360360,23616552960,
%U 596208601546720632677647440,12252240,24240072441867520569208380462960,232792560,232792560,232792560,4860817599682675053132316060135142981520
%N a(n) = least multiple of n such that the geometric mean of a(1), ..., a(n) is an integer.
%C a(11), if it exists, is greater than 10^12. - _Ryan Propper_, Oct 10 2005
%C 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:
%C [2, 1]
%C [2, 2]
%C [2, 1; 3, 3]
%C [2, 5; 3, 1]
%C [2, 2; 3, 1; 5, 5]
%C [2, 2; 3, 1; 5, 1]
%C [2, 2; 3, 1; 5, 1; 7, 7]
%C [2, 10; 3, 1; 5, 1; 7, 1]
%C [2, 3; 3, 10; 5, 1; 7, 1]
%C [2, 3; 3, 2; 5, 1; 7, 1]
%C [2, 3; 3, 2; 5, 1; 7, 1; 11, 11]
%C [2, 3; 3, 2; 5, 1; 7, 1; 11, 1]
%C [2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 13, 13]
%C [2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]
%C [2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]
%C [2, 19; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]
%C [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 17]
%C [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
%C [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 19]
%C [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1]
%C [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1]
%C [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1]
%C [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1; 23, 23]."
%e (1*4*54*96)^(1/4) = (20736)^(1/4) = 12.
%e a(5) = 37500 = 2^2 * 3 * 5^5.
%e a(11) = 718985409939720 = 2^3 * 3^2 * 5 * 7 * 11^11.
%t 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 *)
%Y Cf. A095209, A095211.
%K nonn
%O 1,2
%A _Amarnath Murthy_, Jun 08 2004
%E More terms from _Ryan Propper_, Oct 10 2005
%E a(11) onwards from Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 25 2005