A095210 -id:A095210 - cp's OEIS frontend

A095209 a(0) = 1, and for n > 0, a(n) = the least multiple of prime(n) such that the geometric mean of a(0) to a(n) is an integer.

1, 4, 54, 3750, 504210, 372027810, 144949074270, 209481995953230, 164735296593157290, 401824316553919068810, 2721846739094340967339230, 5095936579799734140259818030, 48850362989361131638352534231610

Offset: 0

Amarnath Murthy, Jun 08 2004

nonn

			(1*4*54*3750)^(1/4) = 30.

Christian Krause, Jamie Morken, et al., A mined LODA assembly source for this sequence
Don Reble, Python program

Cf. A000040, A002110, A020522, A057335, A095210, A307539.

{1}~Join~MapIndexed[Prime[First[#2]]^First[#2]*#1 &, FoldList[Times, Prime@ Range[12]]] (* Michael De Vlieger, Jul 01 2022 *)

A002110(n)=prod(i=1, n, prime(i));
A095209(n) = if(0==n, 1, prime(n)^(n)*A002110(n)); \\ Antti Karttunen, Jun 28 2022

From Antti Karttunen and Peter Munn, May 04 2022: (Start)
The n-th partial product of these terms = A002110(n)^(1+n), i.e., the n-th geometric mean is the n-th power of (n-1)-th primorial.
a(n) = A002110(n) * A307539(n).
a(n) = A057335(A020522(n)). [Found by LODA-miner, follows from the above formulas]
(End)

Edited by Don Reble, Jan 06 2007

Starting offset changed from 1 to 0 and the definition accordingly edited by Antti Karttunen, May 04 2022

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

A095209 a(0) = 1, and for n > 0, a(n) = the least multiple of prime(n) such that the geometric mean of a(0) to 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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