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 A078437 #63 May 06 2025 17:12:10
%S A078437 2,6,9,6,0,6,3,5,1,9,7,1,6,7
%N A078437 Decimal expansion of sum of alternating series of reciprocals of primes.
%C A078437 Verified and extended by _Chris K. Caldwell_ and _Jud McCranie_.
%C A078437 Next two terms are most likely 4 and 5. - _Robert Price_, Sep 13 2011
%C A078437 From _Jon E. Schoenfield_, Nov 25 2018: (Start)
%C A078437 Let f(k) be the k-th partial sum of the alternating series, i.e., f(k) = Sum_{j=1..k} ((-1)^(j+1))/prime(j). At large values of k, successive first differences f(k) - f(k-1) = ((-1)^(k+1))/prime(k) are alternatingly positive and negative and are nearly the same in absolute value, so f(k) is alternatingly above (for odd k) or below (for even k) the value of the much smoother function g(k) = (f(k-1) + f(k))/2 (a two-point moving average of the function f()).
%C A078437 Additionally, since the first differences f(k) - f(k-1) are decreasing in absolute value, g(k) will be less than both g(k-1) and g(k+1) for odd k, and greater than both for even k; i.e., g(), although much smoother than f(), is also alternatingly below or above the value of the still smoother function h(k) = (g(k-1) + g(k))/2 = ((f(k-2) + f(k-1))/2 + (f(k-1) + f(k))/2)/2 = (f(k-2) + 2*f(k-1) + f(k))/4. Evaluated at k = 2^m for m = 1, 2, 3, ..., the values of h(k) converge fairly quickly toward the limit of the alternating series:
%C A078437 h(k) =
%C A078437 k (f(k-2) + 2*f(k-1) + f(k))/4
%C A078437 ========== ============================
%C A078437 2 0.29166666666666666...
%C A078437 4 0.28095238095238095...
%C A078437 8 0.26875529011751921...
%C A078437 16 0.27058892362329746...
%C A078437 32 0.27009944617052797...
%C A078437 64 0.26963971020080367...
%C A078437 128 0.26959147218377685...
%C A078437 256 0.26959653902072193...
%C A078437 512 0.26960402179695026...
%C A078437 1024 0.26960568606633210...
%C A078437 2048 0.26960649673621509...
%C A078437 4096 0.26960645080540929...
%C A078437 8192 0.26960627432070023...
%C A078437 16384 0.26960633643086948...
%C A078437 32768 0.26960634835658329...
%C A078437 65536 0.26960635083481533...
%C A078437 131072 0.26960635144743392...
%C A078437 262144 0.26960635199009778...
%C A078437 524288 0.26960635199971603...
%C A078437 1048576 0.26960635195886861...
%C A078437 2097152 0.26960635197214933...
%C A078437 4194304 0.26960635197019215...
%C A078437 8388608 0.26960635197186919...
%C A078437 16777216 0.26960635197171149...
%C A078437 33554432 0.26960635197146884...
%C A078437 67108864 0.26960635197167534...
%C A078437 134217728 0.26960635197167145...
%C A078437 268435456 0.26960635197166927...
%C A078437 536870912 0.26960635197167200...
%C A078437 1073741824 0.26960635197167416...
%C A078437 2147483648 0.26960635197167454...
%C A078437 4294967296 0.26960635197167462... (End)
%C A078437 The above mentioned average functions can also be written g(k) = f(k) + (-1)^k/prime(k)/2 and h(k) = g(k) + (-1)^k (1/prime(k) - 1/prime(k-1))/4 = f(k) + (-1)^k (3/prime(k) - 1/prime(k-1))/4. - _M. F. Hasler_, Feb 20 2024
%D A078437 S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98.
%H A078437 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>
%H A078437 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>
%H A078437 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_zeta_function">Prime Zeta Function</a>
%F A078437 c = lim_{n -> oo} A024530(n)/A002110(n). - _M. F. Hasler_, Feb 20 2024
%e A078437 1/2 - 1/3 + 1/5 - 1/7 + 1/11 - 1/13 + ... = 0.26960635197167...
%t A078437 s = NSum[ p=Prime[k//Round]; (-1)^k/p, {k, 1, Infinity}, WorkingPrecision -> 30, NSumTerms -> 5*10^7, Method -> "AlternatingSigns"]; RealDigits[s, 10, 14] // First (* _Jean-François Alcover_, Sep 02 2015 *)
%o A078437 (PARI) L=2^N=1; h=List([1/4, 1/6 + S=.5-1/o=3]); forprime(p=o+1,oo, S+=(-1)^L/p; L--|| print([L=2^N++, p, S, listput(h, S+(3/p-1/o)/4)]); o=p) \\ in PARI version > 2.13 listput() may not return the element so one must add +h[#h]
%o A078437 A(x,n=#x)=(x[n]*x[n-2]-x[n-1]^2)/(x[n]+x[n-2]-2*x[n-1]) \\ This is Aitken's Delta-square extrapolation for the last 3 elements of the list x. One can check that the extrapolation is useful for the sequence of raw partial sums (f(2^k)), but not for the smooth/average sequence (h(2^k)). - _M. F. Hasler_, Feb 20 2024
%Y A078437 Cf. A242301, A242302, A242303, A242304.
%Y A078437 Cf. A024530 (numerator of partial sums), A002110 (denominators: primorials).
%K A078437 cons,hard,more,nonn
%O A078437 0,1
%A A078437 _G. L. Honaker, Jr._, Dec 31 2002
%E A078437 Values of a(11)-a(14) = 7,1,6,7 from _Robert Price_, Sep 13 2011