A324072 - OEIS

%I #10 Feb 20 2019 15:09:56

%S 35,143,209,247,323,527,589,713,851,899,989,1073,1147,1247,1333,1591,

%T 1763,2257,2479,2501,2623,2747,2867,2881,2993,3139,3149,3233,3239,

%U 3397,3431,3551,3599,3713,3869,3953,4087,4187,4307,4453,4661,4693,4819,4891,5141,5183

%N For any composite number k take the polynomial defined by the product of the terms (x-d_i), where d_i are the aliquot parts of k. Integrate this polynomial from the minimum to the maximum value of d_i. Sequence lists the numbers k for which the integral is a positive integer.

%C Composites with an integral equal to zero are listed in A129521.

%C Similar to A203612 where prime factors are taken into account.

%e Aliquot parts of 35 are 1, 5, 7. Polynomial: (x-1)*(x-5)*(x-7) = x^3 - 13*x^2 + 47*x - 35. Integral: x^4/4 - (13/3)*x^3 + (47/2)*x^2 - 35*x. The area from x=1 to x=7 is 36.

%p with(numtheory): P:=proc(n) local a,k,x,y;

%p a:=sort([op(divisors(n) minus {n})]);

%p y:=int(mul((x-k),k=a),x=1..a[nops(a)]);

%p if frac(y)=0 and y>0 then n; fi; end: seq(P(i),i=2..5183);

%Y Cf. A129521, A203612, A203613, A203614, A245284, A324073.

%K nonn,easy

%O 1,1

%A _Paolo P. Lava_, Feb 14 2019