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

1,3

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

Links

Table of n, a(n) for n=1..61.

Wikipedia, Smooth number

Example

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A105732 A339328 A337378 * A271783 A114415 A320141

Adjacent sequences: A166316 A166317 A166318 * A166320 A166321 A166322

Keyword

nonn

Author

Claudio Meller, Oct 11 2009

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]

Status

approved