login
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.
7
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
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
Sequence in context: A334558 A320712 A338344 * A250330 A034278 A116870
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Jan 05 2012
STATUS
approved