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

Author

Michel Lagneau, Nov 24 2010

Keywords

• nonn
• base

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,...

Examples

			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.
		

Programs

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:

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