login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A160910 Decimal expansion of c = sum over twin primes (p, p+2) of (1/p^2 + 1/(p+2)^2). 7
2, 3, 7, 2, 5, 1, 7, 7, 6, 5, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Compare Viggo Brun's constant (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + (1/29 + 1/31) + ... (see A065421, A005597).
It appears that c = Sum 1/A001359(n)^2 + 1/A006512(n)^2. - R. J. Mathar, May 30 2009
0.237251776574746 < c < 0.237251776947124. - Farideh Firoozbakht, May 31 2009
c < 0.2725177657771. - Hagen von Eitzen, Jun 03 2009
From Farideh Firoozbakht, Jun 01 2009: (Start)
We can show that a(9)=6, a(10)=5, and a(11) is in the set {7, 8, 9}.
Proof: s1 = 0.237251776576249072... is the sum up to prime(499,000,000) and s2 = 0.237251776576250009... is the sum up to prime(500,000,000).
By using the fact that number of twin primes between the first 10^6*n primes and the first 10^6*(n+1) primes is decreasing (up to the first 2*10^9 primes), we conclude that the sum up to prime(2,000,000,000) is less than s2 + 1500*(s2-s1).
But since s2-s1 < 10^(-15), the sum up to prime(2*10^9) is less than s2 + 1.5*10^(-12) = 0.237251776576250009... + 1.5*10^(-12) = 0.237251776577550009... .
Hence the constant c is less than
0.237251776577550009... + lim(sum(1/k^2,{k, prime(2,000,000,001), n}, n -> infinity)
< 0.237251776577550009... + 2.12514*10^(-11)
< 0.237251776598801409.
So we have 0.237251776576250009 < c < 0.237251776598801409, hence a(9)=6, a(10)=5, and a(11) is in the set {7, 8, 9}.
I guess that a(11)=7. (End)
From Jon E. Schoenfield, Jan 02 2019: (Start)
Given that the Hardy-Littlewood approximation to the number of twin prime pairs < y is
2 * C_2 * Integral_{x=2..y} dx/log(x)^2
where C_2 = 0.660161815846869573927812110014555778432623 (see A152051), we can estimate the size of the tail of the summation Sum(1/A001359(j)^2) + 1/A006512(j)^2) for twin primes > y as
t(y) = 2 * C_2 * Integral_{x>y} 2*dx/(x*log(x))^2.
Let s(y) be the sum of the squares of the reciprocals of all the twin primes <= y, and let s'(y) = s(y) + t(y) be the result of adding to the actual value s(y) the estimated tail size t(y). Evaluating s(y), t(y), and s'(y) at y = 2^d for d = 20..33 gives
.
d s(2^d) t(2^d)*10^10 s(2^d) + t(2^d)
== ==================== ============ ====================
20 0.237251764919808326 115.34589710 0.237251776454398036
21 0.237251771317612979 52.59702970 0.237251776577315949
22 0.237251774173347724 24.08221952 0.237251776581569676
23 0.237251775469086555 11.06766714 0.237251776575853269
24 0.237251776066813995 5.10395459 0.237251776577209454
25 0.237251776340760021 2.36119196 0.237251776576879217
26 0.237251776467109357 1.09553336 0.237251776576662693
27 0.237251776525743797 0.50967952 0.237251776576711749
28 0.237251776552887645 0.23771866 0.237251776576659511
29 0.237251776565549906 0.11113468 0.237251776576663374
30 0.237251776571456873 0.05207020 0.237251776576663892
31 0.237251776574218065 0.02444677 0.237251776576662742
32 0.237251776575513036 0.01149984 0.237251776576663020
33 0.237251776576121140 0.00541938 0.237251776576663078
.
which agrees with all the terms in the Data section and suggests likely values for additional terms.
(End)
LINKS
EXAMPLE
(1/9 + 1/25) + (1/25 + 1/49) + (1/121 + 1/169) + (1/289 + 1/361) + (1/841 + 1/961) + ... = 0.237251...
CROSSREFS
Sequence in context: A105273 A174925 A204986 * A292389 A195306 A330421
KEYWORD
nonn,cons,more
AUTHOR
William Royle (seriesandsequences(AT)yahoo.com), May 29 2009
EXTENSIONS
R. J. Mathar pointed out that the value of c as originally submitted was incorrect (see link). - N. J. A. Sloane, May 31 2009
More terms from Farideh Firoozbakht and Hagen von Eitzen, Jun 01 2009
Name changed by Michael B. Porter, Jan 04 2019
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)