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

1,1

Comments

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

Similar to A203612 where prime factors are taken into account.

Links

Table of n, a(n) for n=1..46.

Example

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.

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);

Crossrefs

Cf. A129521, A203612, A203613, A203614, A245284, A324073.

Sequence in context: A220014 A157286 A354543 * A327901 A346910 A290560

Adjacent sequences: A324069 A324070 A324071 * A324073 A324074 A324075

Keyword

nonn,easy

Author

Paolo P. Lava, Feb 14 2019

Status

approved