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A245284
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For any composite number n with more than a single prime factor, take the polynomial defined by the product of the terms (x-pi)^ei, where pi are the prime factors of n with multiplicities ei. Integrate this polynomial from the minimum to the maximum value of pi. This sequence lists the numbers for which the integral is an integer.
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4
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55, 85, 91, 105, 115, 133, 140, 145, 187, 195, 204, 205, 217, 231, 235, 247, 253, 259, 265, 275, 285, 295, 301, 319, 351, 355, 357, 385, 391, 403, 415, 425, 427, 429, 445, 451, 465, 469, 476, 481, 483, 493, 505, 511, 517, 535, 553, 555, 559, 565, 575, 583, 589
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OFFSET
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1,1
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COMMENTS
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LINKS
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EXAMPLE
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n=1001. Prime factors: 7, 11 and 13: min(pi)=7, max(pi)=13. Polynomial: (x-7)*(x-11)*(x-13)= x^3-31*x^2+311*x-1001. Integral: x^4/4-31/3*x^3+311/2*x^2-1001*x. The area from x=7 to x=13 is 36.
n=1005. Prime factors: 3, 5 and 67: min(pi)=3, max(pi)=67. Polynomial: (x-3)*(x-5)*(x-67)= x^3-75*x^2+551*x-1005. Integral: x^4/4-25*x^3+551/2*x^2-1005*x. The area from x=3 to x=67 is -1310720.
n=1470. Prime factors: 2, 3, 5 and 7^2: min(pi)=2, max(pi)=7. Polynomial: (x-2)*(x-3)*(x-5)*(x-7)^2= x^5-24*x^4+220*x^3-954*x^2+1939*x-1470. Integral: x^6/6-24/5*x^5+55*x^4-318*x^3+1939/2*x^2-1470*x. The area from x=3 to x=67 is 0.
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MAPLE
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isA245284 := proc(n)
local pfs, x1, x2, po, x ;
if isprime(n) then
false;
else
pfs := ifactors(n)[2] ;
if nops(pfs) > 1 then
po := mul((x-op(1, p))^op(2, p), p=pfs) ;
int(po, x=x1..x2) ;
type(%, 'integer') ;
else
false;
end if;
end if;
end proc:
for n from 4 to 600 do
if isA245284(n) then
printf("%d, ", n) ;
end if;
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Definition and example corrected by R. J. Mathar, Sep 07 2014
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STATUS
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approved
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