A284154 Number of zeros in the left half-plane of the polynomial whose coefficients are the ordered divisors of n.

0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 1, 3, 1, 5, 1, 3, 1, 1, 1, 4, 1, 1, 1, 5, 1, 3, 1, 3, 3, 1, 1, 5, 2, 3, 1, 3, 1, 3, 1, 5, 1, 1, 1, 7, 1, 1, 3, 4, 1, 3, 1, 3, 1, 5, 1, 7, 1, 1, 3, 3, 1, 3, 1, 5, 2, 1, 1, 7, 1, 1, 1, 3, 1, 7

Offset

1,4

Comments

Let {d_1= 1, d_2, ..., d_q= n} be the ordered list of the q divisors of n. a(n) is the number of zeros in the left half-plane of the polynomial P(n,X) = Sum_{k=1..q} d_k * X^(k-1).

We observe that a(n) = A084115(n) except for n = 24, 30, 40, 48, 56, 60, 64, 70, 72, 80, 84, 90, ...

Links

Table of n, a(n) for n=1..90.

Example

a(12) = 3 because the divisors of 12 are 1, 2, 3, 4, 6 and 12, hence P(12,X) = 1 + 2*X + 3*X^2 + 4*X^3 + 6*X^4 + 12*X^5, and the zeros are:
X1= -0.5711989847...,
X2= -0.2975767212... - 0.4961486201...*i,
X3= -0.2975767212... + 0.4961486201...*i,
X4=  0.3331762136... - 0.5699669416...*i,
X5=  0.3331762136... + 0.5699669416...*i, where i = sqrt(-1).
There are three zeros X1, X2 and X3 in the left half-plane.

Maple

with(numtheory): for n from 1 to 90 do:it:=0:
d:=divisors(n):n0:=nops(d):P:=add(op(i, d)*x^(i-1), i=1..n0):y:=[fsolve(P, x, complex)]:for m from 1 to nops(y) do:if Re(y[m])<0 then it:=it+1:else fi:od: printf(`%d, `, it):od:

Crossrefs

Cf. A084115.

Sequence in context: A083896 A332083 A084115 * A080028 A309228 A309778

Adjacent sequences: A284151 A284152 A284153 * A284155 A284156 A284157

Keyword

nonn

Author

Michel Lagneau, Mar 21 2017

Status

approved