[VA] SRC #012c - Then and Now: Sum

[J-F Garnier]

Since we are several to have tried the brute force approach, here are my results that show that it's a no go.

I rewrote 1/f(n) to 1/(n.f(d)) and used the fact that all the n values between 2^d and 2^(d+1)-1 have the same number of binary digits d+1 so the sum of these values can be factored with the same f(d+1).
Translated into code (sorry for the change of notation):
J1=2^(K-1) @ J2=2*J1-1
S=0 @ FOR J=J1 TO J2 @ S=S+1/J @ NEXT J
S1=S1+S/F(K)

I also noticed that the intermediate value S converges to LOG(2), so to have an idea of how the series behaves I replace the partial sum S with LOG(2) for K>=15, i.e. for the f(n) with n>2^15.
I went up to K=4000, i.e. numbers with 4000 binary digits (> 10^1200) , and the sum has still not converged.
I can just say that the sum is larger than 1.95, which is compatible with both Werner and Thomas prelim results.

Any confirmation of my analysis is welcome!

10 ! SRC12C
20 L=4000
30 DIM F(L)
40 ! calculate the first L F(I)
50 F(1)=1 @ F(2)=2
60 FOR I=3 TO L
70 D=INT(LOG(I)/LOG(2))+1
90 F(I)=I*F(D)
100 NEXT I
120 !
130 S1=1/F(1)+1/F(2)+1/F(3)
140 FOR K=3 TO L
142 S=LOG(2) ! approx value used for K>=15
145 IF K>=15 THEN 210
150 J1=2^(K-1) @ J2=2*J1-1
170 S=0 @ FOR J=J1 TO J2 @ S=S+1/J @ NEXT J
210 S1=S1+S/F(K)
220 IF K<15 OR MOD(K,500)=0 THEN DISP K;S;S1
230 NEXT K
235 END

>RUN
K, partial sum S, sum S1
3 .759523809524 1.79325396826
4 .725371850372 1.82347779536
5 .709016202207 1.84711166877
6 .701020708269 1.86658446622
7 .697068688885 1.88318133976
8 .695104120223 1.88680167372
9 .694124696722 1.89001521398
10 .693635700225 1.89290536273
11 .693391380792 1.89553184523
12 .693269265777 1.89793903018
13 .69320821942 1.9001608514
14 .693177699089 1.90222388027
500 .69314718056 1.95019974933 (S estimated from here)
1000 .69314718056 1.95220716995
1500 .69314718056 1.95327727197
2000 .69314718056 1.95403237901
2500 .69314718056 1.95457439294
3000 .69314718056 1.9550131171
3500 .69314718056 1.95538406366
4000 .69314718056 1.95570539887

J-F