A324073 - OEIS

%I #7 Feb 20 2019 15:10:05

%S 14,21,26,32,33,38,39,49,51,57,62,65,69,74,86,87,93,95,111,122,123,

%T 125,129,133,134,141,146,155,158,159,169,177,182,183,185,194,201,206,

%U 213,215,217,218,219,237,242,249,254,259,267,273,278,291,301,302,303,305

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

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

%C Similar to A203613 where prime factors are taken into account.

%C If all the divisors were considered, then prime numbers with an integral with a negative integer would be those listed in A002476.

%e Aliquot parts of 32 are 1, 2, 4, 8, 16. Polynomial: (x-1)*(x-2)*(x-4)*(x-8)*(x-16) = x^5-31*x^4+310*x^3-1240*x^2+1984*x-1024. Integral: x^6/6-31/5*x^5+155/2*x^4-1240*x^3/3+992*x^2-1024*x. The area from x=1 to x=16 is -81000.

%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..305);

%Y Cf. A002476, A129521, A203612, A203613, A203614, A245284, A324072.

%K nonn,easy

%O 1,1

%A _Paolo P. Lava_, Feb 14 2019