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A087980 Numbers with strictly decreasing prime exponents. 12
1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 72, 96, 128, 144, 192, 256, 288, 360, 384, 432, 512, 576, 720, 768, 864, 1024, 1152, 1440, 1536, 1728, 2048, 2160, 2304, 2592, 2880, 3072, 3456, 4096, 4320, 4608, 5184, 5760, 6144, 6912, 8192, 8640, 9216, 10368, 10800 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This representation provides a natural ordering between strictly decreasing sequences of natural numbers. Let f and g be such sequences with f(1) > f(2) > ... > f(m) and g(1) > g(2) > ... > g(n). Define f < g iff p^f < p^g, where p^f is short for Product(i=1..m) p_i^f(i) and p^g is defined likewise as Product(i=1..n) p_i^g(i).

Note that "strictly decreasing sequences of natural numbers" is another way to say "partitions into distinct parts".

Also products of primorial numbers p_1#^k_1 * p_2#^k_2 * ... * p_n#^k_n where all k_i > 0.

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

FORMULA

The numbers of the form Product(i=1..n) p_i^k_i where p_i = A000040(i) is the i-th prime and k_1 > k_2 > ... > k_n are positive natural numbers.

Compute x = 2^k_1 * 3^k_2 * 5^k_3 * 7^k_4 * 11^k_5 for k_1 > ... > k_5 allowing k_i = 0 for i > 1 and k_i = k_(i+1) in that case. Discard all x > 174636000 = 2^5*3^4*5^3*7^2*11 and enumerate those below. For more members take higher primes into account.

EXAMPLE

The sequence starts with a(1)=1, a(2)=2, a(3)=4 and a(4)=8. The next term is a(5)=12 = 2^2*3^1 = p_1^k_1 * p_2^k_2 with k_1=2 > k_2=1.

CROSSREFS

Cf. A000040, A025487, A002110, A000009.

Sequence in context: A070173 A116882 A069519 * A181818 A170892 A052184

Adjacent sequences:  A087977 A087978 A087979 * A087981 A087982 A087983

KEYWORD

easy,nice,nonn

AUTHOR

Rainer Rosenthal, Oct 27 2003

EXTENSIONS

Edited by Franklin T. Adams-Watters, Apr 25 2006

Offset change to 1 by T. D. Noe, May 24 2010

STATUS

approved

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Last modified April 20 01:48 EDT 2014. Contains 240777 sequences.