|
| |
|
|
A137245
|
|
Decimal expansion of sum 1/(p * log p) over the primes p=2, 3, 5, 7,...
|
|
7
|
|
|
|
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 [math.NT], 2008-2009, 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} 1/(A000040(n)*log A000040(n)).
|
|
|
EXAMPLE
|
1.63661632335...
|
|
|
CROSSREFS
|
Cf. A221711 (p squared), A115563, A319231 (log squared), A319232 (p and log squared).
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
|
| |
|
|
