A348126 a(n) is the least k for which x(k) > Sum_{j=1..k} 1/j, where x(k) is the Newton iterate (starting from x(1)=1) for exp(-P(n,x)) and P(n,x) is the n-th Maclaurin polynomial for exp(x).

2, 3, 11, 36, 104, 287, 776, 2074, 5519, 14672, 38999, 103709, 275970, 734862, 1958108, 5220797, 13927895, 37176046, 99277815, 265238573, 708928640, 1895558566, 5070252360

Offset

1,1

Comments

Conjecture: a(n) is finite for every positive integer n, and a(n) approaches infinity with n.

Links

Table of n, a(n) for n=1..23.

Example

Let H(k) = Sum_{j=1..k} 1/j, i.e., the k-th harmonic number.
a(1)=2: Newton's method applied to f(x)=exp(-1-x) starting at x(1)=1 gives x(1)=1, x(2)=2, whereas H(1)=1 and H(2)=1.5.
a(3)=11: Newton's method applied to f(x) = exp(-1 - x - x^2/2 - x^3/6) starting at x(1)=1 gives x(10)=2.91631 and x(11)=3.03873, whereas H(10)=2.92897 and H(11)=3.01988.

Prog

(PARI)
a(n)=
{default(seriesprecision, n);
default(realprecision, max(38, 2*#digits(n!, 10)));
Tp = truncate(taylor(exp(x), x, n));
t=1.0;
hsum=1.0;
j=1;
while(t<=hsum, j=j+1; t=t+1.0/subst(Tp, x, t); hsum = hsum + 1.0/j);
j; }

Crossrefs

Cf. A001008, A002805.

Sequence in context: A305846 A057838 A219497 * A006497 A038912 A019361

Adjacent sequences: A348123 A348124 A348125 * A348127 A348128 A348129

Keyword

nonn,more

Author

Jennifer Sirkin, Oct 01 2021

Extensions

a(15) inserted by Jinyuan Wang, Dec 10 2021

Status

approved