login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A137245 Decimal expansion of sum 1/(p * log p) over the primes p=2,3,5,7,11,... 5
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, A115563.

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

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

License Agreements, Terms of Use, Privacy Policy .

Last modified December 2 13:03 EST 2016. Contains 278678 sequences.