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A203614 For any number n take the polynomial formed by the product of the terms (x-pi), where pi’s are the prime factors of n. Then calculate the area between the minimum and the maximum value of the prime factors. This sequence lists the numbers for which the area is equal to zero.

%I A203614 #12 Mar 30 2012 18:53:50
%S A203614 105,140,231,627,748,750,897,935,1470,1581,1729,2205,2465,2625,2967,
%T A203614 3404,3549,4123,4301,4715,5452,5487,6256,7623,7685,7881,9009,9717,
%U A203614 10707,10829,10988,11319,11339,13310,14993,15470,16377,17353,17457,17901,20213,20915
%N A203614 For any number n take the polynomial formed by the product of the terms (x-pi), where pi’s are the prime factors of n. Then calculate the area between the minimum and the maximum value of the prime factors. This sequence lists the numbers for which the area is equal to zero.
%C A203614 Prime numbers are excluded because are banal solutions: in fact for them min(pi)=max(pi)=pi and then the area is zero.
%C A203614 Any squarefree number with an odd number of prime factors which are symmetrically distributed around the central one is part of the sequence. For instance with n=53295 the prime factors are 3, 5, 11, 17, 19 and 3+8=11=19-8, 5+6=11=17-6.
%H A203614 Paolo P. Lava, <a href="/A203614/b203614.txt">Table of n, a(n) for n = 1..1000</a>
%e A203614 n=140. Prime factors: 2, 2, 5, 7: min(pi)=2, max(pi)=7. Polynomial to integrate from 2 to 7: (x-2)^2*(x-5)*(x-7)=x^4-16*x^3+87*x^2-188x+140. The resulting area is equal to zero.
%p A203614 with(numtheory);
%p A203614 P:=proc(i)
%p A203614 local a,b,c,d,k,m,m1,m2,n;
%p A203614 for k from 1 to i do
%p A203614 a:=ifactors(k)[2]; b:=nops(a); c:=op(a); d:=1;
%p A203614 if b>1 then
%p A203614    m1:=c[1,1]; m2:=0;
%p A203614    for n from 1 to b do
%p A203614      for m from 1 to c[n][2] do d:=d*(x-c[n][1]); od;
%p A203614      if c[n,1]<m1 then m1:=c[n,1]; fi;  if c[n,1]>m2 then m2:=c[n,1]; fi;
%p A203614    od;
%p A203614    if int(d,x=m1..m2)=0 then print(k); fi;
%p A203614 fi;
%p A203614 od;
%p A203614 end:
%p A203614 P(500000);
%Y A203614 Cf. A203612, A203613.
%K A203614 nonn
%O A203614 1,1
%A A203614 _Paolo P. Lava_, Jan 05 2012