0,2

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 priomorial numbers p_1#^k_1 * p_2#^k_2 * ... * p_n#^k_n where all k_i > 0.

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.

The sequence starts with a(0)=1, a(1)=2, a(2)=4 and a(3)=8. The next term is a(4)=12 = 2^2*3^1 = p_1^k_1 * p_2^k_2 with k_1=2 > k_2=1.

Cf. A000040, A025487, A002110, A000009.

Sequence in context: A070173 A116882 A069519 this_sequence A052184 A152768 A160686

Adjacent sequences: A087977 A087978 A087979 this_sequence A087981 A087982 A087983

easy,nice,nonn

Rainer Rosenthal (r.rosenthal(AT)web.de), Oct 27 2003

Edited by Frank Adams-Watters (FrankTAW(AT)Netscape.net), Apr 25 2006