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.

429, 605, 663, 969, 1001, 1105, 1183, 1311, 1445, 1653, 1955, 2139, 2185, 2261, 2527, 2553, 2645, 2697, 2755, 3179, 3219, 3335, 3741, 3813, 4199, 4205, 4371, 4551, 4693, 4807, 4929, 4991, 5217, 5289, 5819, 5865, 5883, 5945, 5957, 6063, 6293, 6355, 6549, 6630

Offset: 1

Author

Paolo P. Lava, Jan 05 2012

Keywords

• nonn

Examples

			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.
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.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A203613, A203614.

Programs

Maple

with(numtheory);
P:=proc(i)
local a,b,c,d,k,m,m1,m2,n,p;
for k from 1 to i do
a:=ifactors(k)[2]; b:=nops(a); c:=op(a); d:=1;
if b>1 then
   m1:=c[1,1]; m2:=0;
   for n from 1 to b do
     for m from 1 to c[n][2] do d:=d*(x-c[n][1]); od;
     if c[n,1]m2 then m2:=c[n,1]; fi;
   od;
   p:=int(d,x=m1..m2); if (trunc(p)=p and p>0) then print(k); fi;
fi;
od;
end:
P(500000);

Mathematica

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 *)