login
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.
7
55, 85, 91, 115, 133, 145, 187, 195, 204, 205, 217, 235, 247, 253, 259, 265, 275, 285, 295, 301, 319, 351, 355, 357, 385, 391, 403, 415, 425, 427, 445, 451, 465, 469, 476, 481, 483, 493, 505, 511, 517, 535, 553, 555, 559, 565, 575, 583, 589, 595, 609, 621, 637
OFFSET
1,1
LINKS
EXAMPLE
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.
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.
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);
CROSSREFS
Sequence in context: A140377 A065912 A245284 * A366848 A320507 A039533
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Jan 05 2012
STATUS
approved