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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). 0
2, 3, 11, 36, 104, 287, 776, 2074, 5519, 14672, 38999, 103709, 275970, 734862, 1958108, 5220797, 13927895, 37176046, 99277815, 265238573, 708928640, 1895558566, 5070252360 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 5 19:48 EST 2023. Contains 360087 sequences. (Running on oeis4.)