A076514 a(1)=1, a(n) is the smallest integer > a(n-1) such that the continued fraction for 1/a(1)+1/a(2)+....+1/a(n) contains strictly more elements than the continued fraction for 1/a(1)+1/a(2)+....+1/a(n-1).

1, 2, 3, 11, 16, 17, 19, 20, 21, 24, 25, 27, 29, 31, 32, 37, 39, 71, 81, 82, 89, 94, 97, 98, 99, 101, 103, 106, 109, 115, 116, 124, 163, 171, 187, 227, 251, 252, 298, 346, 353, 359, 394, 424, 438, 452, 509, 542, 590, 643, 677, 685, 751, 810, 882, 1063, 1123

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..205

Example

Continued fraction for 1+1/2+1/3+1/11+1/16 is [1, 1, 74, 2, 3] which contains 5 elements, the continued fraction for 1+1/2+1/3+1/11+1/16+1/17 is [2, 21, 1, 17, 1, 1, 2, 4] which contains 8 elements, hence a(6)=17

Maple

with(NumberTheory):
R:= 1: S:= 1: m:= 1: r:= 1:
for n from 2 to 100 do
  for k from r+1 do
    C:= ContinuedFraction(S+1/k);
    v:= nops(Term(C, all));
    if v > m then
       R:= R, k; m:= v; S:= S+1/k; r:=k; break
    fi
  od
od:
R; # Robert Israel, Jan 30 2025

Prog

(PARI) a(n)=if(n<0, 0, s=a(n-1)+1; while(length(contfrac(1/s+sum(i=1, n-1, 1/a(i))))<=length(contfrac(sum(i=1, n-1, 1/a(i)))), s++); s)

Crossrefs

Sequence in context: A066687 A144979 A194558 * A071012 A354742 A228520

Adjacent sequences: A076511 A076512 A076513 * A076515 A076516 A076517

Keyword

nonn

Author

Benoit Cloitre, Nov 09 2002

Status

approved