A087980 Numbers with strictly decreasing prime exponents.

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

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

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

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.

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]];
Select[Range[12000], selQ] (* Jean-François Alcover, Mar 27 2020, after first PARI program *)

Prog

(Haskell)
import Data.List (isPrefixOf)
a087980 n = a087980_list !! (n-1)
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
(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.

Cf. A000009, A000040, A002110, A027748, A055932, A124010, A133813, A317791, A328524, A357864.

Sequence in context: A116882 A069519 A363948 * A181818 A363063 A336496

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