OFFSET
0,1
COMMENTS
From Robert Price, Jul 14 2010: (Start)
This series converges very slowly. I could not find any transform that converges faster, so I did this by brute force using 256 bits of precision.
After k=596765000000 terms (p(k)=17581469834441) the partial sum is 0.848 969 034 043 245 206 069 544 346 415 327 714...
The next two digits are either 29 or 30. (End)
The table in the Example section shows, for increasing values of j, the results of computing the partial sum s(j) = Sum_{k=1..j} 1/(k*prime(k)) and adding to it an approximate value for the tail (i.e., the sum for all the terms k > j). See the Links entry for an explanation of the method used in approximating the size of the tail of the summation beyond the j-th prime. - Jon E. Schoenfield, Jan 20 2019
LINKS
Jon E. Schoenfield, Notes on approximating the size of the summation's tail beyond the j-th prime
Eric Weisstein's World of Mathematics, Prime Number Theorem
EXAMPLE
0.848969034043...
From Jon E. Schoenfield, Jan 14 2019: (Start)
We can obtain prime(2^d) for d = 0..57 from the b-file for A033844. Given the above result from Robert Price, and letting j_RP = 596765000000, the partial sum through
prime(j_RP) = 17581469834441
is
s(j_RP) = Sum_{k=1..j_RP} 1/(k*prime(k))
= 0.848969034043245206069544346415327714...;
adding to this actual partial sum s(j_RP) the approximate tail value
t(j_RP) =
h'(prime(j_RP), prime(2^40))
+ (Sum_{d=41..57} h'(prime(2^(d-1)), prime(2^d)))
+ lim_{x->infinity} h(prime(2^57), x)
(see the Links entry for an explanation) gives the result 0.84896903404330021273712255895762255... (which seems likely to be correct to at least 20 significant digits).
The table below gives, for j = 2^16, 2^17, ..., 2^32, and j_RP, the actual partial sum s(j) and the sum s(j) + t(j) where t(j) is the approximate tail value beyond prime(j).
.
j s(j) s(j) + t(j)
==== ====================== ======================
2^16 0.84896790758922908159 0.84896903393397518971
2^17 0.84896850050492294891 0.84896903400552099072
2^18 0.84896878057566843770 0.84896903404214147367
2^19 0.84896891330602605081 0.84896903404317536927
2^20 0.84896897639243509768 0.84896903404350431035
2^21 0.84896900645590169648 0.84896903404376063663
2^22 0.84896902081581006534 0.84896903404343742139
2^23 0.84896902768965496764 0.84896903404337393698
2^24 0.84896903098637626311 0.84896903404331189996
2^25 0.84896903257029535468 0.84896903404329806633
2^26 0.84896903333252861584 0.84896903404330030271
2^27 0.84896903369988697984 0.84896903404330084536
2^28 0.84896903387717904236 0.84896903404330042023
2^29 0.84896903396285181513 0.84896903404330024036
2^30 0.84896903400430044877 0.84896903404330021861
2^31 0.84896903402437548991 0.84896903404330021472
2^32 0.84896903403410856545 0.84896903404330021655
... ... ...
j_RP 0.84896903404324520607 0.84896903404330021274
(End)
CROSSREFS
KEYWORD
AUTHOR
Pierre CAMI, Nov 02 2006
EXTENSIONS
Offset and leading zero corrected by R. J. Mathar, Jan 31 2009
Four more terms (4,0,4,3) from Robert Price, Jul 14 2010
Title and example edited by M. F. Hasler, Jan 13 2015
STATUS
approved