A097319 - OEIS

%I #24 Feb 23 2024 07:28:32

%S 18,50,54,75,98,108,147,162,242,245,250,324,338,363,375,486,500,507,

%T 578,605,648,686,722,845,847,867,972,1029,1058,1083,1125,1183,1250,

%U 1372,1445,1458,1587,1682,1715,1805,1859,1875,1922,1944,2023,2250

%N Numbers with more than one prime factor and, in the ordered factorization, the exponents are strictly increasing.

%C If n = Product[k=1..m, p(k)^e(k)], then m>1 and e(1) < e(2) <...< e(m).

%H T. D. Noe, <a href="/A097319/b097319.txt">Table of n, a(n) for n = 1..1180</a>

%e 507 is 3^1*13^2, A001221(507)=2 and 1<2, so 507 is in sequence.

%e 150 is 2^1*3^1*5^2 is not in the sequence because 1,1,2 is not strictly increasing (although it is nondecreasing).

%t fQ[n_] := Module[{d, f = FactorInteger[n]}, If[Length[f] == 1, False, d = Differences[Transpose[f][[2]]]; And @@ ((# > 0) & /@ d)]]; Select[Range[2250], fQ] (* _T. D. Noe_, Apr 09 2013 *)

%o (PARI) for(n=1, 3000, F=factor(n); t=0; s=matsize(F)[1]; if(s>1, for(k=1, s-1, if(F[k, 2]>=F[k+1, 2], t=1; break)); if(!t, print1(n", "))))

%o (PARI) is(n) = my(f = factor(n)[,2]); #f > 1 && vecsort(f, , 8) == f \\ _Rick L. Shepherd_, Jan 17 2018

%Y Subset of A126706. Cf. A097318, A097320.

%K nonn

%O 1,1

%A _Ralf Stephan_, Aug 04 2004