|
%I #24 Sep 07 2014 17:21:04
%S 55,85,91,105,115,133,140,145,187,195,204,205,217,231,235,247,253,259,
%T 265,275,285,295,301,319,351,355,357,385,391,403,415,425,427,429,445,
%U 451,465,469,476,481,483,493,505,511,517,535,553,555,559,565,575,583,589
%N 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.
%C The union of A203612 U A203613 U A203614.
%H Paolo P. Lava, <a href="/A245284/b245284.txt">Table of n, a(n) for n = 1..1000</a>
%e 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.
%e 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.
%e 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.
%p isA245284 := proc(n)
%p local pfs,x1,x2,po,x ;
%p if isprime(n) then
%p false;
%p else
%p pfs := ifactors(n)[2] ;
%p if nops(pfs) > 1 then
%p x1 := A020639(n) ;
%p x2 := A006530(n) ;
%p po := mul((x-op(1,p))^op(2,p),p=pfs) ;
%p int(po,x=x1..x2) ;
%p type(%,'integer') ;
%p else
%p false;
%p end if;
%p end if;
%p end proc:
%p for n from 4 to 600 do
%p if isA245284(n) then
%p printf("%d,",n) ;
%p end if;
%p end do: # _R. J. Mathar_, Sep 07 2014
%Y Cf. A203612, A203613, A203614, A245435. Subsequence of A024619.
%K nonn,easy
%O 1,1
%A _Paolo P. Lava_, Aug 22 2014
%E Definition and example corrected by _R. J. Mathar_, Sep 07 2014
|
