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A245284 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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The union of A203612 U A203613 U A203614.

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..1000

EXAMPLE

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.

MAPLE

isA245284 := proc(n)

    local pfs, x1, x2, po, x ;

    if isprime(n) then

        false;

    else

        pfs := ifactors(n)[2] ;

        if nops(pfs) > 1 then

            x1 := A020639(n) ;

            x2 := A006530(n) ;

            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;

end do: # R. J. Mathar, Sep 07 2014

CROSSREFS

Cf. A203612, A203613, A203614, A245435. Subsequence of A024619.

Sequence in context: A135984 A140377 A065912 * A203613 A320507 A039533

Adjacent sequences:  A245281 A245282 A245283 * A245285 A245286 A245287

KEYWORD

nonn,easy

AUTHOR

Paolo P. Lava, Aug 22 2014

EXTENSIONS

Definition and example corrected by R. J. Mathar, Sep 07 2014

STATUS

approved

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Last modified August 23 11:47 EDT 2019. Contains 326222 sequences. (Running on oeis4.)