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A085892 Group the natural numbers such that the product of the n-th 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 (list; graph; refs; listen; history; text; internal format)



It appears that it is always possible to achieve exactly n prime factors in the n-th group, but a proof would be nice. - Franklin T. Adams-Watters, Sep 07 2006

Empirically, the n-th 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. Adams-Watters, Sep 07 2006


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


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


Cf. A085893, A085894.

Sequence in context: A276451 A075352 A116655 * A101177 A117625 A297763

Adjacent sequences: A085889 A085890 A085891 * A085893 A085894 A085895




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


More terms from Ray Chandler, Sep 13 2003



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Last modified February 6 11:14 EST 2023. Contains 360104 sequences. (Running on oeis4.)