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.

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

Offset: 1

Author

Paolo P. Lava, Aug 22 2014

Keywords

• nonn
• easy

Comments

The union of A203612 U A203613 U A203614.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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

Extensions

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