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

1,1

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.

Links

Table of n, a(n) for n=1..56.

Example

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.

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);

Crossrefs

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

Sequence in context: A224771 A096017 A274226 * A006614 A039832 A108633

Adjacent sequences: A324070 A324071 A324072 * A324074 A324075 A324076

Keyword

nonn,easy

Author

Paolo P. Lava, Feb 14 2019

Status

approved