A203613 - OEIS

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.

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

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OFFSET

1,1

LINKS

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

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

Cf. A203612, A203614.

Sequence in context: A140377 A065912 A245284 * A366848 A320507 A039533

Adjacent sequences: A203610 A203611 A203612 * A203614 A203615 A203616

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Jan 05 2012

STATUS

approved