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A284154
Number of zeros in the left half-plane of the polynomial whose coefficients are the ordered divisors of n.
0
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, ...
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 21 2017
STATUS
approved