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A137245 Decimal expansion of sum 1/(p * log p) over the primes p=2,3,5,7,11,... 4
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, 6, 7, 1, 1, 8, 1, 5, 9, 7, 0, 7, 6, 1, 3, 1, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

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]

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]

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

  (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]

LINKS

Table of n, a(n) for n=1..50.

H. Cohen, High-precision calculation of Hardy-Littlewood constants, (1998).

P. Erdos, Note on sequences of integers no one of which is divisible by any other, J. London Math. Soc. 10 (1935), pp. 126-128, [DOI].

P. Erdos, Some problems and results on combinatorial number theory, 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.

R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739, table in Section 2.4.

Re: What is Sum(1/p log p)?, posting by David C. Ullrich on sci.tech-archive.net (Feb 2006).

FORMULA

Equals sum_(n=1,2,..,infinity} 1/(A000040(n)*log A000040(n)).

EXAMPLE

1.63661632335...

CROSSREFS

Cf. A221711.

Sequence in context: A176715 A229522 A227400 * A060294 A181171 A193025

Adjacent sequences:  A137242 A137243 A137244 * A137246 A137247 A137248

KEYWORD

cons,nonn

AUTHOR

R. J. Mathar, Mar 09 2008

STATUS

approved

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Last modified April 19 05:22 EDT 2014. Contains 240738 sequences.