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

1,1

Links

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

Example

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.

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]<m1 then m1:=c[n, 1]; fi; 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 *)

Crossrefs

Cf. A203613, A203614.

Sequence in context: A334558 A320712 A338344 * A250330 A034278 A116870

Adjacent sequences: A203609 A203610 A203611 * A203613 A203614 A203615

Keyword

nonn

Author

Paolo P. Lava, Jan 05 2012

Status

approved