A324072 For any composite number k take the polynomial defined by the product of the terms (x-d_i), where d_i are the aliquot parts of k. Integrate this polynomial from the minimum to the maximum value of d_i. Sequence lists the numbers k for which the integral is a positive integer.
35, 143, 209, 247, 323, 527, 589, 713, 851, 899, 989, 1073, 1147, 1247, 1333, 1591, 1763, 2257, 2479, 2501, 2623, 2747, 2867, 2881, 2993, 3139, 3149, 3233, 3239, 3397, 3431, 3551, 3599, 3713, 3869, 3953, 4087, 4187, 4307, 4453, 4661, 4693, 4819, 4891, 5141, 5183
Offset: 1
Examples
Aliquot parts of 35 are 1, 5, 7. Polynomial: (x-1)*(x-5)*(x-7) = x^3 - 13*x^2 + 47*x - 35. Integral: x^4/4 - (13/3)*x^3 + (47/2)*x^2 - 35*x. The area from x=1 to x=7 is 36.
Programs
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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..5183);
Comments