A324210 Squarefree numbers k such that the sum of the distinct prime factors of k is twice the difference between the largest and the smallest prime factors of k.

110, 182, 374, 494, 782, 1334, 2294, 3182, 3854, 4982, 6254, 7905, 7917, 8174, 9782, 11534, 12765, 14774, 15810, 15834, 18705, 19982, 20757, 21614, 22330, 22454, 24182, 25530, 27265, 28210, 30381, 30597, 32637, 35894, 37410, 40205, 41181, 41514, 43005, 47414, 49210

Offset: 1

Author

David A. Corneth, Apr 09 2019

Keywords

• nonn

Comments

This sequence is a primitive subsequence of A200070. If p|a(n) for some prime p then p*a(n) is in A200070.

From Robert Israel, Apr 09 2019: (Start)

All terms have at least three prime factors.

The number of prime factors is odd if and only if the term is even.

The terms with three prime factors are 2*A111192. (End)

Examples

			110 = 2 * 5 * 11 is squarefree. The minimal and maximal prime divisors of 110 are 2 and 11 respectively. Twice their difference is 2 * (11-2) = 18 which is also the sum of the distinct prime divisors of 110; 2 + 5 + 11 = 18.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A200070, A111192.

Programs

Maple

filter:= proc(n) local P;
if not numtheory:-issqrfree(n) then return false fi;
P:= numtheory:-factorset(n);
  convert(P, `+`) = 2*(max(P)-min(P))
end proc:
select(filter, [$1..50000]);# Robert Israel, Apr 09 2019

Mathematica

Select[Select[Range[2, 5*10^4], SquareFreeQ], Total@ # == 2 (Last@ # - First@ #) &@ FactorInteger[#][[All, 1]] &] (* Michael De Vlieger, Apr 11 2019 *)

PARI

is(n) = if(!issquarefree(n), return(0)); my(f=factor(n)[, 1]~); sum(i=1, #f, f[i])==2*(f[#f]-f[1])
forcomposite(c=1, 50000, if(is(c), print1(c, ", "))) \\ Felix Fröhlich, Apr 11 2019