A095209 -id:A095209 - cp's OEIS frontend

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

Amarnath Murthy, Jun 08 2004

nonn

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]."

			(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.

Cf. A095209, A095211.

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 *)

More terms from Ryan Propper, Oct 10 2005

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

A095211 Lexicographically least sequence such that a(n) is a positive multiple of the n-th composite number and the arithmetic mean of the first n terms is an integer.

Original entry on oeis.org

4, 6, 8, 162, 30, 84, 14, 60, 64, 18, 540, 42, 242, 336, 100, 130, 540, 140, 900, 480, 132, 170, 490, 36, 10222, 78, 2360, 84, 5148, 90, 3082, 96, 1568, 50, 5814, 52, 594, 880, 1736, 1938, 348, 360, 1364, 504, 1024, 2730, 5346, 204, 28980, 420, 576, 1628

Offset: 1

Amarnath Murthy, Jun 08 2004

nonn

Cf. A095209, A095210, A095212.

Edited by Don Reble, Mar 17 2007

A095212 a(n) = arithmetic mean of A095211(1 to n).

Original entry on oeis.org

4, 5, 6, 45, 42, 49, 44, 46, 48, 45, 90, 86, 98, 115, 114, 115, 140, 140, 180, 195, 192, 191, 204, 197, 598, 578, 644, 624, 780, 757, 832, 809, 832, 809, 952, 927, 918, 917, 938, 963, 948, 934, 944, 934, 936, 975, 1068, 1050, 1620, 1596, 1576, 1577, 1618

Offset: 1

Amarnath Murthy, Jun 08 2004

nonn

Cf. A095209, A095210, A095211.

Edited by Don Reble, Mar 17 2007

cp's OEIS Frontend

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

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A095211 Lexicographically least sequence such that a(n) is a positive multiple of the n-th composite number and the arithmetic mean of the first n terms is an integer.

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A095212 a(n) = arithmetic mean of A095211(1 to n).

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