A194098 -id:A194098 - cp's OEIS frontend

A356793 Decimal expansion of sum of squares of reciprocals of lesser twin primes, Sum_{j>=1} 1/(A001359(j))^2.

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

Offset: 0

Artur Jasinski, Sep 04 2022

Alternative definition: sum of squares of reciprocals of primes whose distance from the next prime is equal to 2.
Convergence table:
   k       A001359(k)      Sum_{j=1..k} 1/A001359(j)^2
10000000   3285916169 0.165618465394273171950874120818
20000000   7065898967 0.165618465394707600197099741096
30000000  11044807451 0.165618465394836120901019351544
40000000  15151463321 0.165618465394895965582366015390
50000000  19358093939 0.165618465394930089884704869090
60000000  23644223231 0.165618465394951950670948192842
Using the Hardy-Littlewood prediction of the density of twin primes (see A347278), the contribution to the sum after the last entry in the table above can be estimated as 9.056*10^(-14), making the infinite sum ~= 0.16561846539504... . - Hugo Pfoertner, Sep 28 2022

			0.165618465395...

Jeffrey P.S. Lay, Sign changes in Mertens' first and second theorems, arXiv:1505.03589 [math.NT], 2015.
Mark B. Villarino, Mertens' Proof of Mertens' Theorem, arXiv:math/0504289 [math.HO], 2005.
Marek Wolf, Generalized Brun's constants, IFTUWr 910/97 (1998), 1-15.
Marek Wolf, Some heuristics on the gaps between consecutive primes, arXiv:1102.0481 [math.NT]. 2011.

Cf. A006512, A065421, A077800, A078437, A085548, A096247, A160910, A194098, A209328, A209329, A242301, A242302, A242303, A242304, A306539, A342714.

Cf. A347278.

Data extended to ...3, 9, 5 by Hugo Pfoertner, Sep 28 2022

A195099 Inverse permutation of A195098; every positive integer occurs exactly once.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 14, 16, 17, 18, 20, 21, 19, 22, 23, 24, 26, 27, 28, 25, 29, 30, 31, 33, 34, 35, 36, 32, 37, 38, 39, 41, 42, 43, 45, 44, 40, 46, 47, 48, 50, 51, 52, 54, 55, 53, 49, 56, 57, 58, 60, 61, 62, 64, 65, 66, 63, 59, 67, 68, 69

Offset: 1

Clark Kimberling, Sep 08 2011

nonn

Cf. A194098, A195097, A194959.

Mathematica
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(See A195098.)
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A213007 Decimal expansion of Brun's quadruple primes constant.

Original entry on oeis.org

8, 7, 0, 5, 8, 8, 3, 8

Offset: 0

Stanislav Sykora, Jun 01 2012

Infinite sum of the reciprocals of p, p+2, p+6, and p+8, where p ranges over all elements of A007530.

The 9th digit is probably 0.

			0.8705883800[+-5]

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.14, p. 133.

Thomas R. Nicely, Enumeration to 1.6e15 of the prime quadruplets.
Wikipedia, Brun's theorem.
Wikipedia, Prime quadruplet.

Cf. A007530, A194098, A065421.

Sum_{n >= 1} (1/c(n) + 1/(c(n)+2) + 1/(c(n)+6) + 1/(c(n)+8)), where c(n) = A007530(n).

A357059 Decimal expansion of sum of squares of reciprocals of primes whose distance to the next prime is equal to 4, Sum_{j>=1} 1/A029710(j)^2.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Sep 10 2022

Convergence table:
   k       A029710(k)      Sum_{j=1..k} 1/A029710(j)^2
10000000   3285441223 0.031321620645456519799598611681
20000000   7067090263 0.031321620645890982910821292996
30000000  11044597393 0.031321620646019474620358985896
40000000  15153534937 0.031321620646079307404248696076
50000000  19360462153 0.031321620646113421819579063642
60000000  23647877233 0.031321620646135276227114122713
70000000  28000392817 0.031321620646150384406674037099

			0.031321620646...

Cf. A006512, A023200, A029710, A046132, A065421, A077800, A078437, A085548, A096247, A160910, A194098, A209328, A209329, A242301, A242302, A242303, A242304, A306539, A342714, A356793.

aa = {}; Do[g1[2 n] = 0, {n, 1, 1000}]; Do[g2[2 n] = 0, {n, 1, 1000}]; Do[g3[2 n] = 0, {n, 1, 1000}]; Do[g4[2 n] = 0, {n, 1, 1000}]; Do[g[2 n] = 0, {n, 1, 1000}];
w1 = 3; n = 3; Monitor[While[n < 10^10, w2 = NextPrime[w1]; kk = w2 - w1;
  If[kk < 2000, g[kk] = g[kk] + 1; g1[kk] = g1[kk] + N[1/w1, 1000];g2[kk] = g2[kk] + N[1/w1^2, 1000];g3[kk] = g3[kk] + N[1/w1^3, 1000];g4[kk] = g4[kk] + N[1/w1^4, 1000];
If[IntegerQ[g[kk]/1000000], Print[{n, w1, kk, g[kk]}];If[kk == 4,AppendTo[aa, {n, w1, kk, g[kk], g1[kk], g2[kk], g3[kk], g4[kk]}]]]];w1 = w2; n++], n];aa

A277774 Decimal expansion of the prime triples constant, also known as Brun's constant B_{3a} = Sum (1/p + 1/(p+2) + 1/(p+6)) as p runs through the initial members of prime triples A022004.

Original entry on oeis.org

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

Offset: 1

Martin Renner, Oct 29 2016

			1.097851039679...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.14, p. 133.

Thomas R. Nicely, Prime Constellations Research Project (Mar 16 2010).

Cf. A022004, A065421, A194098, A213007, A277775.

Offset corrected by Rick L. Shepherd, Nov 03 2016

A277775 Decimal expansion of the prime triples constant, also known as Brun's constant B_{3b} = Sum (1/p + 1/(p+4) + 1/(p+6)) as p runs through the initial members of prime triples A022005.

Original entry on oeis.org

8, 3, 7, 1, 1, 3, 2, 1, 2, 4, 1, 1

Offset: 0

Martin Renner, Oct 29 2016

			0.837113212411...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.14, p. 133.

Thomas R. Nicely, Prime Constellations Research Project (Mar 16 2010).

Cf. A022005, A065421, A194098, A213007, A277774.

Offset corrected by Rick L. Shepherd, Nov 03 2016

cp's OEIS Frontend

A356793 Decimal expansion of sum of squares of reciprocals of lesser twin primes, Sum_{j>=1} 1/(A001359(j))^2.

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A195099 Inverse permutation of A195098; every positive integer occurs exactly once.

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A213007 Decimal expansion of Brun's quadruple primes constant.

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A357059 Decimal expansion of sum of squares of reciprocals of primes whose distance to the next prime is equal to 4, Sum_{j>=1} 1/A029710(j)^2.

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A277774 Decimal expansion of the prime triples constant, also known as Brun's constant B_{3a} = Sum (1/p + 1/(p+2) + 1/(p+6)) as p runs through the initial members of prime triples A022004.

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Extensions

A277775 Decimal expansion of the prime triples constant, also known as Brun's constant B_{3b} = Sum (1/p + 1/(p+4) + 1/(p+6)) as p runs through the initial members of prime triples A022005.

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