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A203612 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 positive integer.

%I A203612 #16 May 27 2016 10:10:01
%S A203612 429,605,663,969,1001,1105,1183,1311,1445,1653,1955,2139,2185,2261,
%T A203612 2527,2553,2645,2697,2755,3179,3219,3335,3741,3813,4199,4205,4371,
%U A203612 4551,4693,4807,4929,4991,5217,5289,5819,5865,5883,5945,5957,6063,6293,6355,6549,6630
%N A203612 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 positive integer.
%H A203612 Paolo P. Lava, <a href="/A203612/b203612.txt">Table of n, a(n) for n = 1..10000</a>
%e A203612 n=1445. Prime factors: 5, 17, 17: min(pi)=5, max(pi)=17. Polynomial: (x-5)*(x-17)^2=x^3-39*x^2+459*x-1445. Integral: x^4/4-13*x^3+459/2*x^2-1445*x. The area from x=5 to x=17 is 1728.
%e A203612 n=999187. Prime factors: 7, 349, 409: min(pi)=7, max(pi)=409. Polynomial: (x-7)*(x-349)*(x-409)=x^3-765*x^2+148047*x-999187. Integral: x^4/4-255*x^3+148047/2*x^2-999187*x. The area from x=7 to x=409 is 1526672988.
%p A203612 with(numtheory);
%p A203612 P:=proc(i)
%p A203612 local a,b,c,d,k,m,m1,m2,n,p;
%p A203612 for k from 1 to i do
%p A203612 a:=ifactors(k)[2]; b:=nops(a); c:=op(a); d:=1;
%p A203612 if b>1 then
%p A203612    m1:=c[1,1]; m2:=0;
%p A203612    for n from 1 to b do
%p A203612      for m from 1 to c[n][2] do d:=d*(x-c[n][1]); od;
%p A203612      if c[n,1]<m1 then m1:=c[n,1]; fi; if c[n,1]>m2 then m2:=c[n,1]; fi;
%p A203612    od;
%p A203612    p:=int(d,x=m1..m2); if (trunc(p)=p and p>0) then print(k); fi;
%p A203612 fi;
%p A203612 od;
%p A203612 end:
%p A203612 P(500000);
%t A203612 apiQ[n_]:=Module[{f=Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[ n]], in}, in = Integrate[Times@@(x-f),{x,f[[1]],f[[-1]]}];Positive[in] && IntegerQ[ in]]; Select[Range[7000],apiQ] (* _Harvey P. Dale_, May 27 2016 *)
%Y A203612 Cf. A203613, A203614.
%K A203612 nonn
%O A203612 1,1
%A A203612 _Paolo P. Lava_, Jan 05 2012