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Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} 1/((2*k-1)*a(k)) < 1.
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%I #16 Oct 14 2024 00:10:21

%S 2,1,2,3,6,172,137534,106557767317,10018727448950607892211,

%T 218107864753736742334588510315735629277159621,

%U 43040465365773907074907163986022284668974202910116417170603263409796800986397420975160781

%N Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} 1/((2*k-1)*a(k)) < 1.

%C It appears that S(n) = (e(n)-1)/e(n) for all n != 4, where e(n) = A376051(n). Exceptionally, S(4) = (e(4)-2)/e(4).

%C a(15) has 1420 decimal digits, too large for a b-file. - _Robert Israel_, Oct 13 2024

%D Rémy Sigrist and N. J. A. Sloane, Dampening Down a Divergent Series, Manuscript in preparation, September 2024.

%H Robert Israel, <a href="/A376050/b376050.txt">Table of n, a(n) for n = 1..14</a>

%p S:= 1:R:= NULL:

%p for i from 1 to 11 do

%p r:= ceil(1/((2*i-1)*S));

%p if r *(2*i-1) = 1/S then r:= r+1 fi;

%p R:= R,r;

%p S:= S - 1/((2*i-1)*r)

%p od:

%p R; # _Robert Israel_, Oct 13 2024

%Y Cf. A374663, A375516, A375531, A375532, A375781, A375522, A376048, A376049, A376051.

%K nonn,base

%O 1,1

%A _N. J. A. Sloane_, Sep 13 2024