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A191580 Numbers n for which the sum of their prime factors (with repetition) divides the sum of their anti-divisors.

%I A191580 #17 May 16 2019 08:52:51
%S A191580 5,10,40,41,129,135,140,155,182,189,200,204,206,238,375,429,435,441,
%T A191580 455,475,546,564,574,616,625,678,722,744,765,836,856,902,1035,1056,
%U A191580 1170,1188,1272,1296,1344,1518,1650,1764,1806,1918,1925
%N A191580 Numbers n for which the sum of their prime factors (with repetition) divides the sum of their anti-divisors.
%H A191580 Paolo P. Lava, <a href="/A191580/b191580.txt">Table of n, a(n) for n = 1..1000</a>
%e A191580 40-> sum prime factors=2+2+2+5=11; sum anti-divisors=3+9+16+27=55; 55/11=5
%e A191580 129-> sum prime factors=3+43=46; sum anti-divisors=2+6+7+37+86=138; 138/46=3
%p A191580 with(numtheory); P:=proc(i) local a,b,j,k,s,n;
%p A191580 for n from 3 to i do b:=ifactors(n)[2];
%p A191580 s:=add(b[k][1]*b[k][2],k=1..nops(b));
%p A191580 k:=0; j:=n; while j mod 2<>1 do k:=k+1; j:=j/2; od; a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
%p A191580 if type(a/s,integer) then print(n); fi; od; end: P(2000);
%Y A191580 Cf. A001414, A066272, A161917, A191581.
%K A191580 nonn,easy
%O A191580 1,1
%A A191580 _Paolo P. Lava_, Jun 07 2011