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

0, 1, 16, 24, 27, 32, 36, 40, 45, 48, 54, 60, 64, 72, 75, 80, 81, 84, 90, 96, 100, 105, 108, 112, 120, 125, 126, 128, 135, 140, 144, 147, 150, 160, 162, 165, 168, 175, 176, 180, 189, 192, 196, 198, 200, 208, 210, 216, 220, 224, 225, 231, 234, 240, 243, 245, 250, 252, 256, 260, 264

Offset: 1

Author

Claudio Meller, Oct 11 2009

Keywords

• nonn

Comments

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

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

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

Examples

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

Links

• Wikipedia, Smooth number

Programs

Maple

isA166319 := proc(n)
    local ifa;
    ifa := ifactors(n)[2] ;
    return (n >= add( op(2,p)*op(1,p)^2,p=ifa)) ;
end proc:
for n from 0 to 1000 do
    if isA166319(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Nov 27 2015

Mathematica

highlySmoothQ[n_] := (pp = Table[#[[1]], #[[2]]]& /@ FactorInteger[n] // Flatten; Times @@ pp >= pp.pp); Select[Range[0, 300], highlySmoothQ] (* Jean-François Alcover, Feb 02 2018 *)

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

Extensions

Edited by Ralf Stephan, Dec 23 2013

Replaced incorrect definition with alternative supplied by Ralf Stephan, Dec 23 2013 [N. J. A. Sloane, Nov 23 2015]