(* Generating the list of s-values *)
lst={};
For[i=2^6,i<=2^7-1,i++,
If[CoprimeQ[i,2*3*7],lst=Append[lst,i]]
];
Print[lst,"    ",Length[lst]];

(* Computations needed in the proof *)

N[2^6.5]

{(1/9)+(1/21),(1/7)+(1/63)}

Simplify[n^3((1/(n-1))-(1/n)-(1/(n^2)))]

d=(1/(7^2))+(1/(63^2))-(1/(9^2))-(1/(21^2))

N[(2/(7^3))+(2/(63^3))-(1/(9^3))-(1/(21^3))]

Rationalize[0.005]

N[(1/(7^3))+(1/(63^3))-(2/(9^3))-(2/(21^3))]

Rationalize[-0.00004]

N[(1/(200*2^6*d))]

Rationalize[0.015]

N[(1/(25000*2^6*d))]

Rationalize[0.00015]

(* Defining the sequence of s-values *)
s[0]=65;s[1]=67;s[2]=71;s[3]=73;s[4]=79;s[5]=83;s[6]=85;s[7]=89;s[8]=95;s[9]=97;s[10]=101;s[11]=103;s[12]=107;s[13]=109;s[14]=113;s[15]=115;s[16]=121;s[17]=125;s[18]=127;

(* More computations needed in the proof *)

N[Max[Table[((1+3/200)/(1-3/20000))(s[j]/s[j+1])^2,{j,0,17}]]]

N[((1+3/200)/(1-3/20000))(s[18]/(2s[0]))^2]

N[((1-3/20000)/(1+3/200)) ((19*(1/4)*2^(-14))/(2^(-12)))]

N[2^(1/38)]

(* Choosing a concrete sum *)

sum1=((10*2)/63)(∑_(j=0)^18)(1/(s[j]));
N[sum1,20]

y=N[(∑_(j=0)^18)(∑_(i=0)^∞)((1/(9*2^i*s[j]-1))+(1/(21*2^i*s[j]-1))),2000]

N[(∑_(j=0)^18)(∑_(i=0)^∞)((1/(7*2^i*s[j]-1))+(1/(63*2^i*s[j]-1))),20]

sum2=Rationalize[0.0656]

(* Defining the sequence of x-terms *)
x[m_]:=(1/(7*2^(Floor[m/19])*s[Mod[m,19]]-1))+(1/(63*2^(Floor[m/19])*s[Mod[m,19]]-1))-(1/(9*2^(Floor[m/19])*s[Mod[m,19]]-1))-(1/(21*2^(Floor[m/19])*s[Mod[m,19]]-1));
For[i=0,i<=20,i++,
Print[N[x[i]]]
];

(* Computing a concrete sequence *)

a={};
For[m=0,m<=2000,m++,
If[x[m]+(∑_(l=0)^(m-1))(e[l]*x[l])>sum2-y,
e[m]=0;a=Append[a,9*2^(Floor[m/19])*s[Mod[m,19]]];
a=Append[a,21*2^(Floor[m/19])*s[Mod[m,19]]],
e[m]=1;a=Append[a,7*2^(Floor[m/19])*s[Mod[m,19]]];
a=Append[a,63*2^(Floor[m/19])*s[Mod[m,19]]]
]
];

nk=Sort[a][[1;;1000]]

N[Plus@@((1/#)&/@nk),20]

N[sum1-(Plus@@((1/#)&/@nk)),20]

N[Plus@@((1/(#-1))&/@nk),20]

N[sum2-(Plus@@((1/(#-1))&/@nk)),20]