cp's OEIS Frontend

A166319 Numbers that are >= the sum of squares of their prime divisors (with multiplicity).

%I A166319 #41 May 28 2020 05:13:09
%S A166319 0,1,16,24,27,32,36,40,45,48,54,60,64,72,75,80,81,84,90,96,100,105,
%T A166319 108,112,120,125,126,128,135,140,144,147,150,160,162,165,168,175,176,
%U A166319 180,189,192,196,198,200,208,210,216,220,224,225,231,234,240,243,245,250,252,256,260,264
%N A166319 Numbers that are >= the sum of squares of their prime divisors (with multiplicity).
%C A166319 Indices m where A067666(m) <= m. Apparently the equality only holds at m=16 and 27.
%C A166319 Members are highly smooth, i.e., they factor mostly into small prime numbers. What are the asymptotics of the largest prime factor of a(n)? - _Ralf Stephan_, Dec 23 2013
%C A166319 Equality A067666(m) = m also holds for 45*A*B where A = (C^107+D^107)/(C+D), B = (C^109+D^109)/(C+D), C = (sqrt(47)+sqrt(43))/2, D = (sqrt(47)-sqrt(43))/2. Maple confirms A and B are prime. - _Michael R Peake_, Apr 09 2020
%H A166319 Wikipedia, <a href="https://en.wikipedia.org/wiki/Smooth_number">Smooth number</a>
%e A166319 24 = 2*2*2*3 >= 2^2 + 2^2 + 2^2 + 3^2 = 21, so 24 is in the sequence.
%p A166319 isA166319 := proc(n)
%p A166319     local ifa;
%p A166319     ifa := ifactors(n)[2] ;
%p A166319     return (n >= add( op(2,p)*op(1,p)^2,p=ifa)) ;
%p A166319 end proc:
%p A166319 for n from 0 to 1000 do
%p A166319     if isA166319(n) then
%p A166319         printf("%d,",n);
%p A166319     end if;
%p A166319 end do: # _R. J. Mathar_, Nov 27 2015
%t A166319 highlySmoothQ[n_] := (pp = Table[#[[1]], #[[2]]]& /@ FactorInteger[n] // Flatten; Times @@ pp >= pp.pp); Select[Range[0, 300], highlySmoothQ] (* _Jean-François Alcover_, Feb 02 2018 *)
%o A166319 (PARI) is(n)=f=factor(n); n>=sum(i=1,matsize(f)[1],f[i,2]*f[i,1]^2) \\ _Ralf Stephan_, Dec 23 2013
%K A166319 nonn
%O A166319 1,3
%A A166319 _Claudio Meller_, Oct 11 2009
%E A166319 Edited by _Ralf Stephan_, Dec 23 2013
%E A166319 Replaced incorrect definition with alternative supplied by _Ralf Stephan_, Dec 23 2013 [_N. J. A. Sloane_, Nov 23 2015]