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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.
1
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.
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);
KEYWORD
nonn,easy
AUTHOR
Paolo P. Lava, Feb 14 2019
STATUS
approved