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A203613 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 a negative integer.

%I A203613 #10 Mar 30 2012 18:53:49
%S A203613 55,85,91,115,133,145,187,195,204,205,217,235,247,253,259,265,275,285,
%T A203613 295,301,319,351,355,357,385,391,403,415,425,427,445,451,465,469,476,
%U A203613 481,483,493,505,511,517,535,553,555,559,565,575,583,589,595,609,621,637
%N A203613 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 a negative integer.
%H A203613 Paolo P. Lava, <a href="/A203613/b203613.txt">Table of n, a(n) for n = 1..10000</a>
%e A203613 n=217. Prime factors: 7, 31: min(pi)=7, max(pi)=31. Polynomial: (x-7)*(x-31)=x^2-38*x+217. Integral: x^3/3-19*x^2+217*x. The area from x=7 to x=31 is -2304.
%e A203613 n=53151. Prime factors: 3, 7, 2531: min(pi)=3, max(pi)=2531. Polynomial: (x-7)*(x-349)*(x-409)=x^3-2541*x^2+25331*x-53151. Integral: x^4/4-847*x^3+25331/2*x^2-53151*x. The area from x=3 to x=2531 is - 3392739409920.
%p A203613 with(numtheory);
%p A203613 P:=proc(i)
%p A203613 local a,b,c,d,k,m,m1,m2,n,p;
%p A203613 for k from 1 to i do
%p A203613 a:=ifactors(k)[2]; b:=nops(a); c:=op(a); d:=1;
%p A203613 if b>1 then
%p A203613    m1:=c[1,1]; m2:=0;
%p A203613    for n from 1 to b do
%p A203613      for m from 1 to c[n][2] do d:=d*(x-c[n][1]); od;
%p A203613      if c[n,1]<m1 then m1:=c[n,1]; fi; if c[n,1]>m2 then m2:=c[n,1]; fi;
%p A203613    od;
%p A203613    p:=int(d,x=m1..m2); if (trunc(p)=p and p<0) then print(k); fi;
%p A203613 fi;
%p A203613 od;
%p A203613 end:
%p A203613 P(500000);
%Y A203613 Cf. A203612, A203614.
%K A203613 nonn
%O A203613 1,1
%A A203613 _Paolo P. Lava_, Jan 05 2012