cp's OEIS Frontend

A324073 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.

%I A324073 #7 Feb 20 2019 15:10:05
%S A324073 14,21,26,32,33,38,39,49,51,57,62,65,69,74,86,87,93,95,111,122,123,
%T A324073 125,129,133,134,141,146,155,158,159,169,177,182,183,185,194,201,206,
%U A324073 213,215,217,218,219,237,242,249,254,259,267,273,278,291,301,302,303,305
%N A324073 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 A324073 Composite with an integral equal to zero are listed in A129521.
%C A324073 Similar to A203613 where prime factors are taken into account.
%C A324073 If all the divisors were considered, then prime numbers with an integral with a negative integer would be those listed in A002476.
%e A324073 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 A324073 with(numtheory): P:=proc(n) local a,k,x,y;
%p A324073 a:=sort([op(divisors(n) minus {n})]);
%p A324073 y:=int(mul((x-k),k=a),x=1..a[nops(a)]);
%p A324073 if frac(y)=0 and y<0 then n; fi; end: seq(P(i),i=2..305);
%Y A324073 Cf. A002476, A129521, A203612, A203613, A203614, A245284, A324072.
%K A324073 nonn,easy
%O A324073 1,1
%A A324073 _Paolo P. Lava_, Feb 14 2019