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 A306700 Decimal expansion of the constant S_2 = Sum_{j>=1} prime(2*j)!/prime(2*j + 1)!. 7
 0, 5, 1, 6, 6, 6, 6, 2, 2, 8, 8, 4, 2 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The constant S_2 is connected to the gap between the j-th and (j+1)-th primes. Together with the constant S_1 (see A306658), S_2 involves the prime gaps, since twin primes produce the heaviest terms of the summation in comparison to their next and previous addend. On Mar 07 2019, the first 2445000000 prime numbers were used and from Rosser's theorem we obtain: 0.05166662288423 < S_2 < 0.05166662288424 + Sum_{j>=1222500000} 1/((2*j*log(2*j) + log(log(2*j)) - 1) * (2*j*log(2*j) + log(log(2*j)) - 2)) < 0.05166662288424 + 3.22757*10^(-13) < 0.05166662288457. LINKS Wikipedia, Rosser's theorem FORMULA Sum_{j>=1} prime(2*j)!/prime(2*j + 1)! = Sum_{j>=1} 1/(Product{k=prime(2*j) + 1, prime(2*j + 1)} k) = 1/(5*4) + 1/(11*10*9*8) + 1/(17*16*15*14) + ... EXAMPLE S_2 = 0.0516666228842... MATHEMATICA b = 0; Do[f = Prime[Range[n - 999999, n]]; Do[b += N[1/Product[k, {k, f[[i]] + 1, f[[i + 1]]}], 100], {i, 1, 1000000, 2}]; Print[n, ": ", N[b, 100]], {n, 1000001, 100000001, 1000000}]; b PROG (PARI) suminf(j=1, prime(2*j)!/prime(2*j + 1)!) \\ Michel Marcus, Apr 02 2019 CROSSREFS Cf. A000040, A306658 (S_1), A306700, A306780. Sequence in context: A131944 A228475 A296355 * A058651 A164105 A262153 Adjacent sequences:  A306697 A306698 A306699 * A306701 A306702 A306703 KEYWORD cons,nonn,more AUTHOR Marco Ripà, Mar 05 2019 STATUS approved

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Last modified June 24 02:56 EDT 2021. Contains 345415 sequences. (Running on oeis4.)