cp's OEIS Frontend

A097318 Numbers with more than one prime factor and, in the ordered factorization, the exponent never increases when read from left to right.

%I A097318 #22 Nov 11 2019 10:50:22
%S A097318 6,10,12,14,15,20,21,22,24,26,28,30,33,34,35,36,38,39,40,42,44,45,46,
%T A097318 48,51,52,55,56,57,58,60,62,63,65,66,68,69,70,72,74,76,77,78,80,82,84,
%U A097318 85,86,87,88,91,92,93,94,95,96,99,100,102,104,105,106,110,111,112,114
%N A097318 Numbers with more than one prime factor and, in the ordered factorization, the exponent never increases when read from left to right.
%C A097318 If n = Product_{k=1..m} p(k)^e(k), then m > 1, e(1) >= e(2) >= ... >= e(m).
%C A097318 These are numbers whose ordered prime signature is weakly decreasing. Weakly increasing is A304678. Ordered prime signature is A124010. - _Gus Wiseman_, Nov 10 2019
%H A097318 Alois P. Heinz, <a href="/A097318/b097318.txt">Table of n, a(n) for n = 1..20000</a>
%H A097318 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper34/page1.htm">Asymptotic formulas for the distribution of integers of various types</a>, Proc. London Math. Soc. 2, 16 (1917), 112-132.
%e A097318 60 is 2^2*3^1*5^1, A001221(60)=3 and 2>=1>=1, so 60 is in sequence.
%p A097318 q:= n-> (l-> (t-> t>1 and andmap(i-> l[i, 2]>=l[i+1, 2],
%p A097318         [$1..t-1]))(nops(l)))(sort(ifactors(n)[2])):
%p A097318 select(q, [$1..120])[];  # _Alois P. Heinz_, Nov 11 2019
%t A097318 fQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Length[f] > 1 && Max[Differences[f]] <= 0]; Select[Range[2, 200], fQ] (* _T. D. Noe_, Nov 04 2013 *)
%o A097318 (PARI) for(n=1, 130, 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", "))))
%Y A097318 Subset of A024619. Cf. A097319, A097320, A230766.
%Y A097318 Cf. A001222, A025487, A118914, A124010, A304678, A329138, A329142.
%K A097318 nonn
%O A097318 1,1
%A A097318 _Ralf Stephan_, Aug 04 2004