A175830 Number of real-valued zeros of the polynomial whose coefficients are the leading n+1 decimal digits of Euler's constant, A011543(n).
0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2
Offset: 0
Examples
a(0) = 0 because 2 => P(0,x)=2 is a constant and has no real root. a(2) = 2 because 271 => P(2,x) = 1+7x + 2x^2 has 2 real roots. a(13) = 3 because 27182818284590 => P(13,x) = 9x +5x^2 +4x^3 +8x^4 +2x^5 +8x^6 +x^7 +8x^8 +2x^9 +8x^10 +x^11 +7x^12 +2x^13 has 3 real roots, -3.664218401…, -0.7829315178… and 0.
Programs
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Maple
with(numtheory):Digits:=50: T:=array(1..45):for zz from 0 to 43 do:n:=floor(exp(1)*10^zz): for i from 1 to 43 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 + T[9]*z^8 + T[10]*z^9+ T[11]*z^10+ T[12]*z^11 + T[13]*z^12 + T[14]*z^13 + T[15]*z^14+ T[16]*z^15+ T[17]*z^16 + T[18]*z^17 + T[19]*z^18 + T[20]*z^19 + T[21]*z^20 + T[22]*z^21+ T[23]*z^22+ T[24]*z^23 + T[25]*z^24 + T[26]*z^25+ T[27]*z^26+ T[28]*z^27+ T[29]*z^28 + T[30]*z^29 + T[31]*z^30+ T[32]*z^31 + T[33]*z^32 + T[34]*z^33+ T[35]*z^34+ T[36]*z^35 + T[37]*z^36 + T[38]*z^37+ T[39]*z^38 + T[40]*z^39+ T[41]*z^40+ T[42]*z^41 + T[43]*z^42, z, real):x1:=[x]: x2:=nops(x1): printf ( "%d %d %d\n",zz,n,x2):od: ~
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