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A377931

a(n) is the numerator of the third convergent of the continued fraction of n^(1/n).

7, 10, 7, 4, 4, 33, 9, 5, 5, 46, 11, 6, 6, 109, 19, 7, 7, 7, 36, 15, 8, 8, 121, 25, 17, 9, 9, 73, 28, 10, 10, 10, 61, 21, 11, 11, 11, 67, 23, 12, 12, 12, 85, 37, 13, 13, 13, 131, 40, 27, 14, 14, 323, 57, 29, 15, 15, 15, 91, 31, 16, 16, 16, 209, 49, 33, 17, 17, 17

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OFFSET

2,1

LINKS

Table of n, a(n) for n=2..70.

EXAMPLE

7/5, 10/7, 7/5, 4/3, 4/3, 33/25, 9/7, 5/4, 5/4, 46/37, 11/9, 6/5, 6/5, 109/91, ...

n^(1/n)-a(n)/A377932(n) for n=2..12: 0.014214, 0.013678, 0.014214, 0.046396, 0.014673, 0.00046925, 0.011125, 0.026518, 0.0089254, 0.00033198, 0.0078533.

MAPLE

c:= n-> (l-> l[1]+1/(l[2]+1/l[3]))(numtheory[cfrac](n^(1/n), 'quotients')):

a:= n-> numer(c(n)):

seq(a(n), n=2..70); # Alois P. Heinz, Nov 11 2024

MATHEMATICA

a[n_] := Convergents[n^(1/n), 3] // Last // Numerator;

Table[a[n], {n, 2, 70}] (* Jean-François Alcover, Mar 16 2025 *)

PROG

(PARI) cofrnthr(n, nc=2) = my(r=n^(1/n), c=contfrac(r, nc+2), pq=contfracpnqn(c, nc)); pq[1, nc]/pq[2, nc];

a377931(n) = numerator(cofrnthr(n, 3))

CROSSREFS

A377932 are the corresponding denominators.