A005597 -id:A005597 - cp's OEIS frontend

A380139 Prime gaps between 10^m and 10^(m+1), m>=0, sorted first by falling number of occurrences and then by rising gap size, written as an irregular triangle.

2, 1, 4, 4, 6, 2, 8, 6, 4, 2, 10, 8, 12, 14, 18, 20, 6, 2, 4, 10, 12, 8, 14, 18, 16, 22, 24, 20, 30, 28, 26, 34, 32, 36, 6, 2, 4, 12, 10, 8, 18, 14, 16, 20, 22, 24, 30, 28, 26, 36, 32, 34, 40, 38, 42, 52, 44, 50, 46, 54, 58, 48, 56, 60, 62, 64, 72

Offset: 1

Hugo Pfoertner based on an idea by Richard Stephen Donovan, Jan 23 2025

A gap between two primes p1 and p2 is assumed to belong to the range [10^m .. 10^(m+1)[ if 10^m <= (p1+p2)/2 < 10^(m+1). Thus the gap between 7 and 11 is counted in the interval 1 .. 10. Gaps symmetric to 10^k occur for k = 17, 45, ... .

			The triangle begins, with corresponding counts in [...]:
  [2, 1, 1]
   2, 1, 4,
  [7, 7, 6, 1]
   4, 6, 2, 8,
  [37, 32, 27, 16, 14,  8,  7,  1,  1]
    6,  4,  2, 10,  8, 12, 14, 18, 20
  [255, 170, 162, 103, 98, 86, 47, 39, 33, 16, 15, 14, 11,  5,  3,  3,  1,  1]
    6,   2,   4,   10, 12,  8, 14, 18, 16, 22, 24, 20, 30, 28, 26, 34, 32, 36,
  [1641, 1018, 1013, 860, 797, 672, 474, 430, 306, 223, 207, 191, 135, 93, 85, ...]
     6,    2,    4,   12,  10,  8,   18,  14,  16,  20,  22,  24,  30, 28, 26, ...
  [11609, 7040, 6945, 6928, 6163, 4796, 4395, 3749, 2542, 2476, 2164, 1949, ...]
     6,    12,    2,    4,   10,    8,   18,   14,   16,   24,   20,   22,  ...
  6, 12, 2, 4, 10, 18, 8, 14, 24, 16, 30, 20, 22, 28, 26, 36, 42, 34, ...
  6, 12, 4, 2, 10, 18, 8, 14, 24, 30, 16, 20, 22, 28, 26, 36, 42, 34, ...
  6, 12, 10, 4, 2, 18, 8, 14, 24, 30, 16, 20, 22, 28, 36, 26, 42, 34, ...
  6, 12, 18, 10, 2, 4, 8, 24, 30, 14, 20, 16, 22, 36, 28, 26, 42, 34, ...

Hugo Pfoertner, Table of n, a(n) for n = 1..583

Cf. A001223, A028334, A038460, A354604.

Cf. A005597, A173557, A305444 for the asymptotic behavior of gap sizes.

A264736 Decimal expansion of Product_{p prime > 2} 1-1/(p^2-3p+3), a constant related to I. M. Vinogradov's proof of the "ternary" Goldbach conjecture.

Original entry on oeis.org

5, 7, 3, 8, 1, 3, 8, 6, 2, 6, 1, 2, 0, 7, 0, 5, 9, 9, 0, 4, 7, 8, 8, 6, 3, 9, 3, 4, 5, 7, 9, 0, 6, 3, 2, 7, 6, 6, 4, 7, 7, 6, 1, 0, 9, 5, 5, 8, 6, 8, 7, 3, 8, 6, 2, 4, 8, 7, 0, 9, 3, 8, 7, 1, 4, 6, 2, 2, 4, 3, 8, 8, 5, 7, 6, 7, 0, 1, 3, 6, 8, 1, 9, 2, 8, 5, 4, 5, 7, 7, 5, 2, 8, 5, 2, 0, 6, 3, 0

Offset: 0

Jean-François Alcover, Apr 17 2016

			0.5738138626120705990478863934579063276647761095586873862487...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 88.

Eric Weisstein's MathWorld, Goldbach's Conjecture.
Eric Weisstein's MathWorld, Vinogradov's theorem.
Wikipedia, Goldbach's conjecture.
Wikipedia, Vinogradov's theorem.

Cf. A005597, A271951.

$MaxExtraPrecision = 600; digits = 99; terms = 600; P[n_] := PrimeZetaP[n] - 1/2^n; LR = LinearRecurrence[{6, -14, 15, -6}, {0, 0, -2, -9}, terms + 10]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n-1]/(n-1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First

prodeulerrat(1-1/(p^2-3*p+3), 1, 3) \\ Amiram Eldar, Mar 11 2021

Equals A005597 / A271951.

A273556 Decimal expansion of Rosser's constant.

Original entry on oeis.org

8, 3, 2, 4, 2, 9, 0, 6, 5, 6, 6, 1, 9, 4, 5, 2, 7, 8, 0, 3, 0, 8, 0, 5, 9, 4, 3, 5, 3, 1, 4, 6, 5, 5, 7, 5, 0, 4, 5, 4, 4, 5, 3, 1, 8, 0, 7, 7, 4, 1, 7, 0, 5, 3, 2, 4, 0, 8, 9, 3, 9, 9, 1, 2, 9, 6, 0, 3, 4, 7, 0, 7, 1, 3, 9, 4, 8, 1, 1, 4, 2, 4, 2, 1, 9, 1, 6, 2, 7, 2, 2, 5, 0, 4, 6, 3, 8, 1

Offset: 0

Jean-François Alcover, May 25 2016

Named after the American logician and mathematician John Barkley Rosser, Sr. (1907-1989). - Amiram Eldar, Jun 20 2021

			0.832429065661945278030805943531465575045445318077417053240893991296...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood constants, p. 86.

J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., Vol. 6, No. 1 (1962), pp. 64-94, eq. (2.14).
Eric Weisstein's World of Mathematics, Twin Primes Constant.

Cf. A005597, A091724, A246061.

digits = 98; s[n_] := (1/n)*N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 60]; C2 = (175/256)*Product[(Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n) )*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, digits + 60}];
RealDigits[4*C2/Exp[2*EulerGamma], 10, digits] // First

4 * exp(-2*Euler) * prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 17 2021

4*C_2/exp(2*EulerGamma), where C_2 is the twin primes constant.

Equals lim_{x->inf} Product_{2 < p <= x} (1-2/p)*log(x)^2.

A347359 Decimal expansion of Product_{p in A077800} (1 - 1/p).

Original entry on oeis.org

1, 2, 9, 3, 3, 7, 1, 7

Offset: 0

Kenneth H. Hicks, Aug 29 2021

Note that A077800 is the sequence of twin primes with 5 repeated. The sequence of twin primes is A001097.
Related to Brun's constant (A065421) and the twin prime constant (A005597).
It is well known that the product of 1-1/p over all primes p is zero (it is related to the Riemann zeta function). Also the sum of 1/p diverges, whereas the sum of 1/p2 for p2 in the sequence A077800 converges to Brun's constant, regardless of whether there are an infinite number of twin primes or not.  Similarly, the product in the present sequence also converges.
The repeated value of 1/5 is used in the calculation of Brun's constant (A065421) and we follow that convention here. The first two pairs of twin primes are (3,5) and (5,7), so the 4 initial terms in the product are (1-1/3)*(1-1/5)*(1-1/5)*(1-1/7).
This constant converges very slowly, similar to the convergence of Brun's constant.  For example, for all twin primes below 1 billion, the product only reaches the value of 0.1469... Details on the error term in the convergence of the above product will be given in a forthcoming paper.

			0.12933717...

K. Hicks and K. Ward, Series and Product Relations Made from Primes, Pi Mu Epsilon Journal, Vol. 15, No. 3, Fall 2020, pp. 161-169.

Ken Hicks and Kevin Ward, Series and Product Relations Made from Primes, arXiv:2108.03268 [math.NT], 2021.

Cf. A001097, A005597, A065421, A077800.

Offset corrected by N. J. A. Sloane, Sep 20 2021

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A380139 Prime gaps between 10^m and 10^(m+1), m>=0, sorted first by falling number of occurrences and then by rising gap size, written as an irregular triangle.

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A264736 Decimal expansion of Product_{p prime > 2} 1-1/(p^2-3p+3), a constant related to I. M. Vinogradov's proof of the "ternary" Goldbach conjecture.

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A273556 Decimal expansion of Rosser's constant.

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A347359 Decimal expansion of Product_{p in A077800} (1 - 1/p).

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