A306700 -id:A306700 - cp's OEIS frontend

A306658 Decimal expansion of the constant S_1 = Sum_{j>=1} prime(2j - 1)!/prime(2j)!.

3, 6, 7, 5, 5, 5, 4, 4, 2, 0, 1, 9, 2, 6

Offset: 0

Marco Ripà, Mar 03 2019

Derived from an open problem published in 2012, the constant S_1 is connected to the gap between the j-th and (j+1)-th primes.
Together with the constant S_2 (see A306700), S_1 involves the prime gaps, since twin primes produce the heaviest terms of the summation in comparison to their next and previous addend.
On Mar 06 2019, the first 4200000000 prime numbers were used and using Rosser's theorem we get:
0.367555442019264 < S_1 < 0.367555442019265 + Sum_{j>=2100000000} 1/((2*j*log(2*j) + log(log(2*j)) - 1) * (2*j*log(2*j) + log(log(2*j)) - 2)) < 0.367555442019265 + 1/1.94576*10^(-13) < 0.367555442019460.

			S_1 = 0.36755544201926...

M. Ripà, Congetture su interrogativi inediti: tra speculazioni, voli pindarici e riflessioni spicciole, Jun 2012. ISBN 9788863699463

Wikipedia, Rosser's theorem

Cf. A000040, A306700.

a = 0; Do[f = Prime[Range[n - 999999, n]]; Do[a += N[1/Product[k, {k, f[[i]] + 1, f[[i + 1]]}], 100], {i, 1, 1000000, 2}]; Print[n, ": ", N[a, 100]], {n, 1000000, 100000000, 1000000}]; a

suminf(j=1, prime(2*j - 1)!/prime(2*j)!) \\ Michel Marcus, Apr 02 2019

Sum_{j>=1} prime(2*j - 1)!/prime(2*j)! = 1/3 + Sum_{j>=2} 1/(Product{k=prime(2*j - 1) + 1, prime(2*j)} k) = 1/3 + 1/(7*6) + 1/(13*12) + ...

A306744 Decimal expansion of the constant S_1 + S_2 = Sum_{j>=1} prime(j)!/prime(j + 1)!.

Original entry on oeis.org

4, 1, 9, 2, 2, 2, 0, 6, 4, 9, 0, 3

Offset: 0

Marco Ripà, Mar 07 2019

The constant S_1 + S_2 is related to the prime gaps, since twin primes produce the largest terms of the sum compared with neighboring terms.

			S_1 + S_2 = 0.419222064903...

Cf. A000040, A306658 (S_1), A306700 (S_2), A306780.

s = 0; p = 2; q = 3; While[p < 10^10, s = N[s + 1/Times @@ Range[p +1, q], 32]; p = q; q = NextPrime@ q]; Take[ RealDigits[s][[1]], 20] (* Robert G. Wilson v, Mar 23 2019 *)

suminf(j=1, prime(j)!/prime(j + 1)!) \\ Michel Marcus, Apr 02 2019

S_1 + S_2 = Sum_{j>=1} prime(j)!/prime(j + 1)! = Sum_{j>=2} 1/(Product{k=prime(j - 1) + 1, prime(j)} k) = 1/3 + 1/(4*5) + 1/(6*7) + 1/(8*9*10*11) + ...

A306780 Decimal expansion of the constant S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!).

Original entry on oeis.org

3, 1, 5, 8, 8, 8, 8, 1, 9, 3, 5, 0

Offset: 0

Marco Ripà and Aldo Roberto Pessolano, Mar 09 2019

The constant S_1 - S_2 is related to the prime gaps, since twin primes produce the largest terms of the algebraic sum compared with neighboring terms.

			S_1 - S_2 = 0.315888819350...

Cf. A000040, A306658 (S_1), A306700 (S_2), A306744 (S_1 + S_2).

sumalt(j=1, (-1)^(j+1)*(prime(j)!/prime(j + 1)!)) \\ Michel Marcus, Apr 02 2019  \\ Needs default(realprecision, 10^4)  Jinyuan Wang, May 19 2019

S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!) = Sum_{j>=2} (-1)^j/(Product{k=prime(j - 1) + 1, prime(j)} k) = 1/3 - 1/(4*5) + 1/(6*7) - 1/(8*9*10*11) + ...

A307383 Decimal expansion of the constant S_1* + S_2* = Sum_{j>=1} prime((j + 1) - 1)!/prime((j + 2) - 1)!.

Original entry on oeis.org

1, 3, 0, 4, 5, 6, 2, 5, 9, 8, 3, 5

Offset: 0

Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

The constant S_1* + S_2* is related to the prime gaps, since twin primes produce the largest terms of the sum compared with neighboring terms.

On Apr 06 2019, the first 4200000000 prime numbers were used in order to calculate S_1* and S_2* and using Rosser's theorem we get: 0.13045626983537 < S_1* + S_2* < 0.13045626983578.

			S_1* + S_2* = 0.130456269835...

Cf. A000040, A306658 (S_1) A306700 (S_2), A306744 (S_1 + S_2).

S_1* + S_2* = Sum_{j>=1} prime((j + 1) - 1)!/prime((j + 2) - 1)! = Sum_{j>=1} 1/(Product{k=prime(j + 1), prime((j + 2) - 1)} k) = 1/(3*4) + 1/(5*6) + 1/(7*8*9*10) + 1/(11*12) + ...

A307384 Decimal expansion of the constant S_1* = Sum_{j>=1} prime((2j) - 1)!/prime((2j + 1) - 1)!.

Original entry on oeis.org

0, 8, 5, 1, 6, 1, 9, 1, 0, 9, 8, 5

Offset: 0

Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

Together with the constant S_2* and S_1* + S_2* (see A307383), S_1* involves the prime gaps, since twin primes produce the heaviest terms of the summation in comparison to their next and previous addend.

On Apr 06 2019, the first 4200000000 prime numbers were used and using Rosser's theorem we get: 0.08516191098523 < S_1* < 0.08516191098543.

			S_1* = 0.085161910985...

Wikipedia, Rosser's theorem

Cf. A000040, A306658 (S_1) A306700 (S_2), A306744 (S_1 + S_2), A307383 (S_1* + S_2*).

S_1* = Sum_{j>=1} prime(2*j - 1)!/prime((2*j + 1) - 1)! = Sum_{j>=1} 1/(Product{k=prime(2*j), prime(2*j + 1)} k) = 1/(3*4) + 1/(7*8*9*10) + 1/(13*14*15*16) + 1/(19*20*21*22) +...

A307385 Decimal expansion of the constant S_2* = Sum_{j>=1} prime((2j + 1) - 1)!/prime((2j + 2) - 1)!.

Original entry on oeis.org

0, 4, 5, 2, 9, 4, 3, 4, 8, 8, 5, 0

Offset: 0

Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

The constant S_2* is related to the prime gaps, since twin primes produce the largest terms of the sum compared with neighboring terms.

On Apr 06 2019, the first 4200000000 prime numbers were used in order to calculate S_1* and S_2* and using Rosser's theorem we get: 0.04529434885014 < S_1* + S_2* < 0.04529434885035.

			S_2* = 0.045294348850...

Wikipedia, Rosser's theorem

Cf. A000040, A306658 (S_1) A306700 (S_2), A306744 (S_1 + S_2), A307383 (S_1* + S_2*), A307384 (S_1*).

S_2* = Sum_{j>=1} prime((2*j + 1) - 1)!/prime((2*j + 2) - 1)! = Sum_{j>=1} 1/(Product{k=prime(2*j + 1), prime((2*j + 2) - 1)} k) = 1/(5*6) + 1/(11*12) + 1/(17*18) + 1/(23*24*25*26*27*28) +...

cp's OEIS Frontend

A306658 Decimal expansion of the constant S_1 = Sum_{j>=1} prime(2*j - 1)!/prime(2*j)!.

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A306744 Decimal expansion of the constant S_1 + S_2 = Sum_{j>=1} prime(j)!/prime(j + 1)!.

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A306780 Decimal expansion of the constant S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!).

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A307383 Decimal expansion of the constant S_1* + S_2* = Sum_{j>=1} prime((j + 1) - 1)!/prime((j + 2) - 1)!.

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A307384 Decimal expansion of the constant S_1* = Sum_{j>=1} prime((2*j) - 1)!/prime((2*j + 1) - 1)!.

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A307385 Decimal expansion of the constant S_2* = Sum_{j>=1} prime((2*j + 1) - 1)!/prime((2*j + 2) - 1)!.

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A306658 Decimal expansion of the constant S_1 = Sum_{j>=1} prime(2j - 1)!/prime(2j)!.

A307384 Decimal expansion of the constant S_1* = Sum_{j>=1} prime((2j) - 1)!/prime((2j + 1) - 1)!.

A307385 Decimal expansion of the constant S_2* = Sum_{j>=1} prime((2j + 1) - 1)!/prime((2j + 2) - 1)!.