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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

Table of n, a(n) for n=0..12.

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.)