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.
f=0; Do[ p=Prime[n]; f=f + (-1)^(n+1)*1/p; g=Numerator[f] ;If[ PrimeQ[g], Print[ {n, g} ] ], {n, 1, 150} ]
1/2 - 1/3 + 1/5 - 1/7 + 1/11 - 1/13 + ... = 0.26960635197167...
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 *)
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] 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
The first few fractions are 3/4, 31/36, 739/900, 37111/44100, 4446331/5336100, 756766039/901800900, ... = A136370/A061742. - _Petros Hadjicostas_, May 14 2020
Table[Numerator[1 - Sum[(-1)^(k+1)/Prime[k]^2, {k, 1, n}]], {n, 1, 20}]
a(n) = numerator(1 - sum(k=1, n, (-1)^(k+1)/prime(k)^2)); \\ Michel Marcus, May 14 2020
from sympy import prime from fractions import Fraction from itertools import accumulate, count, islice def A136370gen(): yield from map(lambda x: (1-x).numerator, accumulate(Fraction((-1)**(k+1), prime(k)**2) for k in count(1))) print(list(islice(A136370gen(), 14))) # Michael S. Branicky, Jun 26 2022
[Numerator(&+[(-1)^(k+1)/NthPrime(k)^2:k in [1..n]]): n in [1..13]]; // Marius A. Burtea, Aug 26 2019
Table[Numerator[Sum[(-1)^(k+1)/Prime[k]^2, {k, 1, n}]], {n, 1, 20}]
f=0; Do[ p=Prime[n]; f=f + (-1)^(n+1)*1/p^2; g=Numerator[f] ;If[ PrimeQ[g], Print[ {n, g} ] ], {n, 1, 100} ]
f=1; Do[ p=Prime[n]; f=f - (-1)^(n+1)*1/p^2; g=Numerator[f] ;If[ PrimeQ[g], Print[ {n, g} ] ], {n, 1, 100} ]
# uses A136370gen() and imports from A136370 from sympy import isprime def agen(): yield from (k for k, ak in enumerate(A136370gen(), 1) if isprime(ak)) print(list(islice(agen(), 5))) # Michael S. Branicky, Jun 26 2022
f=1; Do[ p=Prime[n]; f=f + (-1)^n*1/p; g=Numerator[f] ;If[ PrimeQ[g], Print[ {n, g} ] ], {n, 1, 60} ]
The first 10 values of A140122(n)/a(n) = -1/4, -1/12, -7/36, -17/180, -209/1260, -25/252, -37/252, -281/2772, -9797/69300, -92711/900900. The 10th term of the sum is (-1/4)+(1/6)-(1/9)+(1/10)-(1/14)+(1/15)-(1/21)+(1/22)-(1/25)+(1/26) = -92711/900900 hence a(10) = 900900. The 20th term of the alternating sum is (-1/4)+(1/6)-(1/9)+(1/10)-(1/14)+(1/15)-(1/21)+(1/22)-(1/25)+(1/26)-(1/33)+(1/34)-(1/35)+(1/38)-(1/39)+(1/46)-(1/49)+(1/51)-(1/55)+(1/57) = -5218865543/46849502700, hence a(20) = 46849502700.
The first 10 values of a(n)/A140123(n) = -1/4, -1/12, -7/36, -17/180, -209/1260, -25/252, -37/252, -281/2772, -9797/69300, -92711/900900. The 10th term of the sum is (-1/4)+(1/6)-(1/9)+(1/10)-(1/14)+(1/15)-(1/21)+(1/22)-(1/25)+(1/26) = -92711/900900 hence a(10) = -(-92711) = 92711. The 20th term of the alternating sum is (-1/4)+(1/6)-(1/9)+(1/10)-(1/14)+(1/15)-(1/21)+(1/22)-(1/25)+(1/26)-(1/33)+(1/34)-(1/35)+(1/38)-(1/39)+(1/46)-(1/49)+(1/51)-(1/55)+(1/57) = -5218865543/46849502700, hence a(20) = 5218865543.
(2*3*5*7)*(1/2 - 1/3 - 1/5 + 1/7) = (210)*(23/210) = 23, so a(4) = 23.
a[n_] := Block[{p = Prime@ Range@ n}, Min@ Abs[{1/p}.Transpose@ Tuples[{-1, 1}, n]] Times @@ p]; Array[a, 16] (* Giovanni Resta, Feb 11 2020 *)
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