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A160910 Decimal expansion of c = (1/9 + 1/25) + (1/25 + 1/49) + (1/121 + 1/169) + (1/289 + 1/361) + (1/841 + 1/961) + ... = 0.237251... 2
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)

LINKS

Table of n, a(n) for n=0..10.

Various authors, On the computation of A160910

CROSSREFS

Sequence in context: A105273 A174925 A204986 * A292389 A195306 A174606

Adjacent sequences:  A160907 A160908 A160909 * A160911 A160912 A160913

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

STATUS

approved

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Last modified November 18 09:17 EST 2017. Contains 294879 sequences.