

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
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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.
A124010(a(n),k+1) < A124010(a(n),k), 1 <= k < A001221(a(n)).  Reinhard Zumkeller, Apr 13 2015


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 ith 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.


PROG

(Haskell)
import Data.List (isPrefixOf)
a087980 n = a087980_list !! (n1)
a087980_list = 1 : filter f [2..] where
f x = isPrefixOf ps a000040_list && all (< 0) (zipWith () (tail es) es)
where ps = a027748_row x; es = a124010_row x
 Reinhard Zumkeller, Apr 13 2015


CROSSREFS

Cf. A000040, A025487, A002110, A000009.
Cf. A027748, A124010, A133808, A133813.
Sequence in context: A070173 A116882 A069519 * A181818 A170892 A246468
Adjacent sequences: A087977 A087978 A087979 * A087981 A087982 A087983


KEYWORD

easy,nice,nonn


AUTHOR

Rainer Rosenthal, Oct 27 2003


EXTENSIONS

Edited by Franklin T. AdamsWatters, Apr 25 2006
Offset change to 1 by T. D. Noe, May 24 2010


STATUS

approved



