A065421 -id:A065421 - cp's OEIS frontend

A213007 Decimal expansion of Brun's quadruple primes constant.

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).

A284203 Number of twin prime (A001097) divisors of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 2, 1, 0, 1, 2, 1, 1, 2, 1, 1, 2, 0, 0, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 0, 2, 2, 0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 2, 2, 0, 1, 1, 1, 0, 2, 2, 1, 2, 1, 0, 2, 2, 0, 2, 0, 2, 1, 0, 1, 2, 1, 1, 2, 1, 1, 3, 0, 1, 1, 1, 2

Offset: 1

Ilya Gutkovskiy, Mar 22 2017

nonn

			--------------------------------------------
| n | divisors of n | twin prime    | a(n) |
|   |               | divisors of n |      |
|------------------------------------------
| 1 | {1}           |      {-}      |  0   |
| 2 | {1, 2}        |      {-}      |  0   |
| 3 | {1, 3}        |      {3}      |  1   |
| 4 | {1, 2, 4}     |      {-}      |  0   |
| 5 | {1, 5}        |      {5}      |  1   |
| 6 | {1, 2, 3, 6}  |      {3}      |  1   |
| 7 | {1, 7}        |      {7}      |  1   |
| 8 | {1, 2, 4, 8}  |      {-}      |  0   |
| 9 | {1, 3, 9}     |      {3}      |  1   |
--------------------------------------------

Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A001097, A001221, A005087, A037074, A062729, A065421, A284599.

Cf. A048599 (positions of records).

nmax = 110; Rest[CoefficientList[Series[Sum[Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Length[Select[Divisors[n], PrimeQ[#] && (PrimeQ[# - 2] || PrimeQ[# + 2]) &]], {n, 110}]

concat([0, 0],Vec(sum(k=1, 110, (isprime(k) && (isprime(k - 2) || isprime(k + 2)))* x^k/(1 - x^k)) + O(x^111))) \\ Indranil Ghosh, Mar 22 2017

a(n) = sumdiv(n, d, isprime(d) && (isprime(d-2) || isprime(d+2))); \\ Amiram Eldar, Jun 03 2024

from sympy import isprime, divisors
print([len([i for i in divisors(n) if isprime(i) and (isprime(i - 2) or isprime(i + 2))]) for n in range(1, 111)]) # Indranil Ghosh, Mar 22 2017

G.f.: Sum_{k>=1} x^A001097(k)/(1 - x^A001097(k)).
a(A062729(n)) = 0. - Ilya Gutkovskiy, Apr 02 2017
From Amiram Eldar, Jun 03 2024: (Start)
a(A048599(n)) = n.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A065421 - 1/5 = 1.7021605... . (End)
Additive with a(p^e) = 1 if p is in A001097, and 0 otherwise. - Amiram Eldar, May 15 2025
a(A037074(n)) = 2. - Michel Marcus, May 15 2025

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

A038124 Beatty sequence for Brun's constant.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 98, 100, 102

Offset: 1

Felice Russo

nonn

Index entries for sequences related to Beatty sequences

a(n)=floor(n*A065421).

Updated reference to Brun's constant - R. J. Mathar, Oct 10 2010

A074042 Numerator of Sum_{k=1..n} 1/A077800(k), denominator=A074043.

Original entry on oeis.org

1, 8, 11, 92, 1117, 15676, 281507, 5603888, 167362597, 5328886012, 222844337147, 9761066934176, 583589647901149, 36052483750271224, 2587390775195626139, 190843701043052923832, 19418598540473717052037

Offset: 1

Reinhard Zumkeller, Aug 13 2002

a(n)/A074043(n) -> Brun's constant (A065421).

Definition corrected by Max Alekseyev, May 10 2009

A074043 Denominator of Sum_{k=1..n} 1/A077800(k), numerator=A074042.

Original entry on oeis.org

3, 15, 15, 105, 1155, 15015, 255255, 4849845, 140645505, 4360010655, 178760436855, 7686698784765, 453515228301135, 27664428926369235, 1964174453772215685, 143384735125371745005, 14481858247662546245505

Offset: 1

Reinhard Zumkeller, Aug 13 2002

For n>2, a(n) = A048599(n-1).

A074042(n)/a(n) -> Brun's constant (A065421).

Essentially the same as A048599.

Edited by Max Alekseyev, May 10 2009

A241560 Decimal expansion of the sum of the reciprocals of the averages of the twin prime pairs.

Original entry on oeis.org

9, 2, 8, 8, 3, 5, 8, 2, 7, 1, 3

Offset: 0

Omar E. Pol, May 07 2014

This constant is due to JJGJJG, see link section.

Denominators are in A014574.

			0.92883582713... = Sum_{k>=1} 1/A014574(k) = 1/4 + 1/6 + 1/12 + 1/18 + 1/30 + 1/42 + ...

JJGJJG, La constante entre primos gemelos, gaussianos, Apr 06 2014.

Cf. A001097, A014574, A065421, A077800.

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

A331369 Decimal expansion of Sum_{(p1, p2) is twin prime pair} 1/p1 + 1/p2 - log(p2/p1).

Original entry on oeis.org

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

Offset: 0

Dimitris Valianatos, May 03 2020

Let (p_k, p_(k+1)) twin prime pair. Then log(p_(k+1)/p_k) < 1/p_k + 1/p_(k+1).
Lim_{k -> oo} 1/p_k + 1/p_(k+1) - log(p_(k+1)/p_k) = 0.
This constant is analogous to Euler-Mascheroni constant for twin primes.

			0.0299817108389062696826...

Cf. A077800, A065421, A001359, A006512, A077761.

p = 3; st = 0.0; forprime(n = 5, 1e11, if(n - p == 2, st += 1/p + 1/n - log(n/p)); p = n); print(st)

Equals Sum_{k >= 1} 1/A001359(k) + 1/A006512(k) - log(A006512(k)/A001359(k)).

cp's OEIS Frontend

A213007 Decimal expansion of Brun's quadruple primes constant.

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A284203 Number of twin prime (A001097) divisors of n.

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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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A038124 Beatty sequence for Brun's constant.

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A074042 Numerator of Sum_{k=1..n} 1/A077800(k), denominator=A074043.

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A074043 Denominator of Sum_{k=1..n} 1/A077800(k), numerator=A074042.

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A241560 Decimal expansion of the sum of the reciprocals of the averages of the twin prime pairs.

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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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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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A331369 Decimal expansion of Sum_{(p1, p2) is twin prime pair} 1/p1 + 1/p2 - log(p2/p1).