This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A124012 #38 Feb 16 2025 08:33:03
%S A124012 8,4,8,9,6,9,0,3,4,0,4,3
%N A124012 Decimal expansion of Sum_{k>=1} 1/(k*prime(k)).
%C A124012 From _Robert Price_, Jul 14 2010: (Start)
%C A124012 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.
%C A124012 After k=596765000000 terms (p(k)=17581469834441) the partial sum is 0.848 969 034 043 245 206 069 544 346 415 327 714...
%C A124012 The next two digits are either 29 or 30. (End)
%C A124012 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
%H A124012 Jon E. Schoenfield, <a href="/A124012/a124012.txt">Notes on approximating the size of the summation's tail beyond the j-th prime</a>
%H A124012 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeNumberTheorem.html">Prime Number Theorem</a>
%e A124012 0.848969034043...
%e A124012 From _Jon E. Schoenfield_, Jan 14 2019: (Start)
%e A124012 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
%e A124012 prime(j_RP) = 17581469834441
%e A124012 is
%e A124012 s(j_RP) = Sum_{k=1..j_RP} 1/(k*prime(k))
%e A124012 = 0.848969034043245206069544346415327714...;
%e A124012 adding to this actual partial sum s(j_RP) the approximate tail value
%e A124012 t(j_RP) =
%e A124012 h'(prime(j_RP), prime(2^40))
%e A124012 + (Sum_{d=41..57} h'(prime(2^(d-1)), prime(2^d)))
%e A124012 + lim_{x->infinity} h(prime(2^57), x)
%e A124012 (see the Links entry for an explanation) gives the result 0.84896903404330021273712255895762255... (which seems likely to be correct to at least 20 significant digits).
%e A124012 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).
%e A124012 .
%e A124012 j s(j) s(j) + t(j)
%e A124012 ==== ====================== ======================
%e A124012 2^16 0.84896790758922908159 0.84896903393397518971
%e A124012 2^17 0.84896850050492294891 0.84896903400552099072
%e A124012 2^18 0.84896878057566843770 0.84896903404214147367
%e A124012 2^19 0.84896891330602605081 0.84896903404317536927
%e A124012 2^20 0.84896897639243509768 0.84896903404350431035
%e A124012 2^21 0.84896900645590169648 0.84896903404376063663
%e A124012 2^22 0.84896902081581006534 0.84896903404343742139
%e A124012 2^23 0.84896902768965496764 0.84896903404337393698
%e A124012 2^24 0.84896903098637626311 0.84896903404331189996
%e A124012 2^25 0.84896903257029535468 0.84896903404329806633
%e A124012 2^26 0.84896903333252861584 0.84896903404330030271
%e A124012 2^27 0.84896903369988697984 0.84896903404330084536
%e A124012 2^28 0.84896903387717904236 0.84896903404330042023
%e A124012 2^29 0.84896903396285181513 0.84896903404330024036
%e A124012 2^30 0.84896903400430044877 0.84896903404330021861
%e A124012 2^31 0.84896903402437548991 0.84896903404330021472
%e A124012 2^32 0.84896903403410856545 0.84896903404330021655
%e A124012 ... ... ...
%e A124012 j_RP 0.84896903404324520607 0.84896903404330021274
%e A124012 (End)
%Y A124012 Cf. A033286, A085548, A209329, A210473.
%K A124012 cons,more,nonn
%O A124012 0,1
%A A124012 _Pierre CAMI_, Nov 02 2006
%E A124012 Offset and leading zero corrected by _R. J. Mathar_, Jan 31 2009
%E A124012 Four more terms (4,0,4,3) from _Robert Price_, Jul 14 2010
%E A124012 Title and example edited by _M. F. Hasler_, Jan 13 2015