

A085892


Group the natural numbers such that the product of the nth group has n prime divisors: (1), (2), (3,4), (5,6,), (7,8,9,10), (11,12,13,14), (15,16,17,18,19,20,21), ... Sequence contains the product of the terms of the groups.


2



1, 2, 12, 30, 5040, 24024, 586051200, 5967561600, 33891580800, 5846743244160, 70758332701056000, 1929327666754295808000, 228609915104317824000, 8755238159153560237363200, 5998865771053625032442880000
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OFFSET

0,2


COMMENTS

It appears that it is always possible to achieve exactly n prime factors in the nth group, but a proof would be nice.  Franklin T. AdamsWatters, Sep 07 2006
Empirically, the nth group has on the order of C*n members (where C >= 1 may not be a constant, but appears to grow slowly); the numbers in that group are then about C*n^2/2. At the end of the group, every prime less than the group size is already present, so the smallest number with two prime factors that are not already present is on the order of (C*n)^2. There are then two ways it might be possible to skip over a value: the apparent growth trend in C could reverse, so that it becomes less than 1/sqrt(2); or there could be an extraordinarily short group.  Franklin T. AdamsWatters, Sep 07 2006


LINKS

Table of n, a(n) for n=0..14.


EXAMPLE

a(5) = 24024= 11*12*13*14 and the 5 prime divisors are 2,3,7,11 and 13.


CROSSREFS

Cf. A085893, A085894.
Sequence in context: A276451 A075352 A116655 * A101177 A117625 A297763
Adjacent sequences: A085889 A085890 A085891 * A085893 A085894 A085895


KEYWORD

nonn


AUTHOR

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 10 2003


EXTENSIONS

More terms from Ray Chandler, Sep 13 2003


STATUS

approved



