A173667 Number of real zeros of the polynomial whose coefficients are the decimal expansion of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0

Offset

1,100

Comments

From the example a(121)=2, P(121,x)=(x+1)^2, it is seen that the roots are counted with multiplicity. For n=0, the polynomial would be P=0 with infinitely many real roots. - M. F. Hasler, Nov 24 2010

a(n) is the number of real zeros of the polynomial P(n,x) = Sum_{k=0..p} d(k)

x^k where d(k) are the digits of the decimal expansion of n = d(p) d(p-1)...d(0), n =0,1,2,...

Links

Table of n, a(n) for n=1..112.

Example

a(121) = 2 because 1+2x+x^2 = 0 has 2 real roots.
a(1597200)=6 because x^6 + 5x^5 + 9x^4+7x^3+2x^2 = x^2 (x+2)(x+1)^3 has 6 real roots.

Maple

with(numtheory):T:=array(1..10000000): x2:=0: printf(`%d, `, x2):for n from
  1 to 300 do: for i from 1 to 8 do: T[i]:=0:od: l:=length(n) : n0:=n:for m from
  1 to l do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v :u: T[m]:=u:od:x:=fsolve(T[1]+
  T[2]*z + T[3]*z^2+ T[4]*z^3+ T[5]*z^4 + T[6]*z^5 + T[7]*z^6 + T[8]*z^7, z, real):x1:=[x]: x2:=nops(x1): printf(`%d, `, x2):od:

Prog

(PARI) A173667(n)=sum(k=1, #n=factor(1.*Pol(eval(Vec(Str(n)))))~, (poldegree(n[1, k])==1)*n[2, k] ) /* factorization over the reals => linear factor for each root. poldegree()==1 could be replaced by poldisc()>=0 */ \\ M. F. Hasler, Nov 25 2010

Crossrefs

Sequence in context: A292241 A352727 A332015 * A241541 A086008 A083910

Adjacent sequences: A173664 A173665 A173666 * A173668 A173669 A173670

Keyword

nonn,base

Author

Michel Lagneau, Nov 24 2010

Status

approved