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Numbers that are >= the sum of squares of their prime divisors (with multiplicity).
2

%I #41 May 28 2020 05:13:09

%S 0,1,16,24,27,32,36,40,45,48,54,60,64,72,75,80,81,84,90,96,100,105,

%T 108,112,120,125,126,128,135,140,144,147,150,160,162,165,168,175,176,

%U 180,189,192,196,198,200,208,210,216,220,224,225,231,234,240,243,245,250,252,256,260,264

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

%C Indices m where A067666(m) <= m. Apparently the equality only holds at m=16 and 27.

%C 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Smooth_number">Smooth number</a>

%e 24 = 2*2*2*3 >= 2^2 + 2^2 + 2^2 + 3^2 = 21, so 24 is in the sequence.

%p isA166319 := proc(n)

%p local ifa;

%p ifa := ifactors(n)[2] ;

%p return (n >= add( op(2,p)*op(1,p)^2,p=ifa)) ;

%p end proc:

%p for n from 0 to 1000 do

%p if isA166319(n) then

%p printf("%d,",n);

%p end if;

%p end do: # _R. J. Mathar_, Nov 27 2015

%t 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 (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 nonn

%O 1,3

%A _Claudio Meller_, Oct 11 2009

%E Edited by _Ralf Stephan_, Dec 23 2013

%E Replaced incorrect definition with alternative supplied by _Ralf Stephan_, Dec 23 2013 [_N. J. A. Sloane_, Nov 23 2015]