

A087980


Numbers with strictly decreasing prime exponents.


23



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.
Numbers whose prime indices cover an initial interval of positive integers with strictly decreasing multiplicities. Intersection of A055932 and A304686. First differs from A181818 in having 72.  Gus Wiseman, Oct 21 2022


LINKS



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.


MATHEMATICA

selQ[k_] := Module[{n = k, e = IntegerExponent[k, 2], t}, n /= 2^e; For[p = 3, True, p = NextPrime[p], t = IntegerExponent[n, p]; If[t == 0, Return[n == 1]]; If[t >= e, Return[False]]; e = t; n /= p^e]];


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
(PARI) is(n)=my(e=valuation(n, 2), t); n>>=e; forprime(p=3, , t=valuation(n, p); if(t==0, return(n==1)); if(t>=e, return(0)); e=t; n/=p^e) \\ Charles R Greathouse IV, Jun 25 2017
(PARI) list(lim)=my(v=[], u=powers(2, logint(lim\=1, 2)), w, p=2, t); forprime(q=3, , w=List(); for(i=1, #u, t=u[i]; for(e=1, valuation(u[i], p)1, t*=q; if(t>lim, break); listput(w, t))); v=concat(v, Vec(u)); if(#w==0, break); u=w; p=q); Set(v) \\ Charles R Greathouse IV, Jun 25 2017


CROSSREFS

The weak (weakly decreasing) version is A025487.
The weak opposite (weakly increasing) version is A133808.
The opposite (strictly increasing) version is A133809.
For strictly decreasing prime signature we have A304686.


KEYWORD

easy,nice,nonn


AUTHOR



EXTENSIONS



STATUS

approved



