%I #15 May 04 2023 17:33:55
%S 1,2,3,4,6,8,10,12,14,16,18,22,30,38,54,70,118,126,134,150,166,182,
%T 198,214,246,278,374,534,598,662,790,854,982,1110,1238,1366,1494,1622,
%U 1878,2006,2134,2390,2902,3158,3670,5462,5974,6486,6998,10070,11094,12118
%N Numbers n where abs(s(n)) produces a new minimum, with s(1) = 1 and s(i) = s(i-1) - sign(s(i-1))*(1/i).
%C Positions of decreasing minima of abs(A203810(i)/A203811(i)).
%H Hugo Pfoertner, <a href="/A203812/b203812.txt">Table of n, a(n) for n = 1..131</a>
%e The first 4 fractions f(i)=A203810(i)/A203811(i) 1/1, 1/2, 1/6, -1/12 have decreasing absolute values. Therefore a(1)=1, a(2)=2, a(3)=3, a(4)=4. 5 is not in the sequence, because f(5)=7/60>1/12, but f(6)=-1/20 gives a(5)=6 because 1/20<1/12.
%e Fractions producing further decreasing absolute values are f(8)=-9/280, f(10)=-53/2520, f(12)=-373/27720, f(14)=-2869/360360, f(16)=-547/144144, f(18)=-1291/2450448, f(22)=-13913/232792560, f(30)=93259013/232908956280.
%o (PARI)
%o s=0; d=2;\
%o for (k=1,12500,if(s>0,s-=1/k,s+=1/k);if(abs(s)<d,d=abs(s);print1(k,", ")))
%o \\ _Hugo Pfoertner_, Nov 14 2017
%Y Cf. A203810, A203811.
%K nonn
%O 1,2
%A _Hugo Pfoertner_ and _Rainer Rosenthal_, Jan 06 2012