A284154 - OEIS

%I #10 Mar 22 2017 03:59:13

%S 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,

%T 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,

%U 1,5,1,7,1,1,3,3,1,3,1,5,2,1,1,7,1,1,1,3,1,7

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

%C 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).

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

%e 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:

%e X1= -0.5711989847...,

%e X2= -0.2975767212... - 0.4961486201...*i,

%e X3= -0.2975767212... + 0.4961486201...*i,

%e X4=  0.3331762136... - 0.5699669416...*i,

%e X5=  0.3331762136... + 0.5699669416...*i, where i = sqrt(-1).

%e There are three zeros X1, X2 and X3 in the left half-plane.

%p with(numtheory): for n from 1 to 90 do:it:=0:

%p 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:

%Y Cf. A084115.

%K nonn

%O 1,4

%A _Michel Lagneau_, Mar 21 2017