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A087980 Numbers with strictly decreasing prime exponents.

%I A087980 #26 Feb 15 2024 13:52:17
%S A087980 1,2,4,8,12,16,24,32,48,64,72,96,128,144,192,256,288,360,384,432,512,
%T A087980 576,720,768,864,1024,1152,1440,1536,1728,2048,2160,2304,2592,2880,
%U A087980 3072,3456,4096,4320,4608,5184,5760,6144,6912,8192,8640,9216,10368,10800
%N A087980 Numbers with strictly decreasing prime exponents.
%C A087980 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).
%C A087980 Note that "strictly decreasing sequences of natural numbers" is another way to say "partitions into distinct parts".
%C A087980 Also products of primorial numbers p_1#^k_1 * p_2#^k_2 * ... * p_n#^k_n where all k_i > 0.
%C A087980 A124010(a(n),k+1) < A124010(a(n),k), 1 <= k < A001221(a(n)). - _Reinhard Zumkeller_, Apr 13 2015
%C A087980 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
%H A087980 T. D. Noe, <a href="/A087980/b087980.txt">Table of n, a(n) for n = 1..1000</a>
%F A087980 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.
%F A087980 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.
%e A087980 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.
%t A087980 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]];
%t A087980 Select[Range[12000], selQ] (* _Jean-François Alcover_, Mar 27 2020, after first PARI program *)
%o A087980 (Haskell)
%o A087980 import Data.List (isPrefixOf)
%o A087980 a087980 n = a087980_list !! (n-1)
%o A087980 a087980_list = 1 : filter f [2..] where
%o A087980    f x = isPrefixOf ps a000040_list && all (< 0) (zipWith (-) (tail es) es)
%o A087980          where ps = a027748_row x; es = a124010_row x
%o A087980 -- _Reinhard Zumkeller_, Apr 13 2015
%o A087980 (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
%o A087980 (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
%Y A087980 The weak (weakly decreasing) version is A025487.
%Y A087980 The weak opposite (weakly increasing) version is A133808.
%Y A087980 The opposite (strictly increasing) version is A133809.
%Y A087980 For strictly decreasing prime signature we have A304686.
%Y A087980 Cf. A000009, A000040, A002110, A027748, A055932, A124010, A133813, A317791, A328524, A357864.
%K A087980 easy,nice,nonn
%O A087980 1,2
%A A087980 _Rainer Rosenthal_, Oct 27 2003
%E A087980 Edited by _Franklin T. Adams-Watters_, Apr 25 2006
%E A087980 Offset change to 1 by _T. D. Noe_, May 24 2010