login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A203614 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 equal to zero. 9
105, 140, 231, 627, 748, 750, 897, 935, 1470, 1581, 1729, 2205, 2465, 2625, 2967, 3404, 3549, 4123, 4301, 4715, 5452, 5487, 6256, 7623, 7685, 7881, 9009, 9717, 10707, 10829, 10988, 11319, 11339, 13310, 14993, 15470, 16377, 17353, 17457, 17901, 20213, 20915 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Prime numbers are excluded because are banal solutions: in fact for them min(pi)=max(pi)=pi and then the area is zero.
Any squarefree number with an odd number of prime factors which are symmetrically distributed around the central one is part of the sequence. For instance with n=53295 the prime factors are 3, 5, 11, 17, 19 and 3+8=11=19-8, 5+6=11=17-6.
LINKS
EXAMPLE
n=140. Prime factors: 2, 2, 5, 7: min(pi)=2, max(pi)=7. Polynomial to integrate from 2 to 7: (x-2)^2*(x-5)*(x-7)=x^4-16*x^3+87*x^2-188x+140. The resulting area is equal to zero.
MAPLE
with(numtheory);
P:=proc(i)
local a, b, c, d, k, m, m1, m2, n;
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;
if int(d, x=m1..m2)=0 then print(k); fi;
fi;
od;
end:
P(500000);
CROSSREFS
Sequence in context: A115935 A069702 A239589 * A252069 A133509 A013590
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Jan 05 2012
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)