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.

14, 21, 26, 32, 33, 38, 39, 49, 51, 57, 62, 65, 69, 74, 86, 87, 93, 95, 111, 122, 123, 125, 129, 133, 134, 141, 146, 155, 158, 159, 169, 177, 182, 183, 185, 194, 201, 206, 213, 215, 217, 218, 219, 237, 242, 249, 254, 259, 267, 273, 278, 291, 301, 302, 303, 305

Offset: 1

Author

Paolo P. Lava, Feb 14 2019

Keywords

• nonn
• easy

Comments

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

Similar to A203613 where prime factors are taken into account.

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

Examples

			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.
		

Crossrefs

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

Programs

Maple

with(numtheory): P:=proc(n) local a,k,x,y;
a:=sort([op(divisors(n) minus {n})]);
y:=int(mul((x-k),k=a),x=1..a[nops(a)]);
if frac(y)=0 and y<0 then n; fi; end: seq(P(i),i=2..305);