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A137245 Decimal expansion of sum 1/(p * log p) over the primes p=2, 3, 5, 7,... 7


%S 1,6,3,6,6,1,6,3,2,3,3,5,1,2,6,0,8,6,8,5,6,9,6,5,8,0,0,3,9,2,1,8,6,3,

%T 6,7,1,1,8,1,5,9,7,0,7,6,1,3,1,2

%N Decimal expansion of sum 1/(p * log p) over the primes p=2, 3, 5, 7,...

%C sum_{p= A000040} 1/(p^s * log p) equals this value here if s=1, equals A221711 if s=2, 0.22120334039... if s=3. See arXiv:0811.4739.

%C Erdos (1935) proved that for any sequence where no term divides another, the sum of 1/(x log x) is at most some constant C. He conjectures (1989) that C can be taken to be this constant 1.636..., that is, the primes maximize this sum. - _Charles R Greathouse IV_, Mar 26 2012

%C Note that sum 1/(p * log p) is almost (a tiny bit less than) 1 + 2/Pi = 1+A060294 = 1.63661977236758... (Why is it so close?) - _Daniel Forgues_, Mar 26 2012

%C Sum 1/(p * log p) is quite close to sum 1/n^2 = Pi^2/6 = 1.644934066...

%C (Cf. David C. Ullrich, "Re: What is Sum(1/p log p)?" for why this is so, and which mentions A115563.) - _Daniel Forgues_, Aug 13 2012

%H H. Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High-precision calculation of Hardy-Littlewood constants</a>, (1998).

%H P. Erdos, <a href="http://www.renyi.hu/~p_erdos/1935-04.pdf">Note on sequences of integers no one of which is divisible by any other</a>, J. London Math. Soc. 10 (1935), pp. 126-128, <a href="http://dx.doi.org/10.1112/jlms/s1-10.1.126">[DOI]</a>.

%H P. Erdos, <a href="http://www.renyi.hu/~p_erdos/1989-35.pdf">Some problems and results on combinatorial number theory</a>, Graph theory and its applications: East and West (Jinan, 1986), Ann. New York Acad. Sci., 576 , pp. 132-145, New York Acad. Sci., New York, 1989.

%H R. J. Mathar, <a href="http://arxiv.org/abs/0811.4739">Twenty digits of some integrals of the prime zeta function</a>, arXiv:0811.4739 [math.NT], 2008-2009, table in Section 2.4.

%H <a href="http://sci.tech-archive.net/Archive/sci.math.research/2006-02/msg00052.html">Re: What is Sum(1/p log p)?</a>, posting by David C. Ullrich on sci.tech-archive.net (Feb 2006).

%F Equals Sum_(n>=1} 1/(A000040(n)*log A000040(n)).

%e 1.63661632335...

%Y Cf. A221711 (p squared), A115563, A319231 (log squared), A319232 (p and log squared).

%K cons,nonn

%O 1,2

%A _R. J. Mathar_, Mar 09 2008

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