

A104141


Decimal expansion of 3/Pi^2.


14



3, 0, 3, 9, 6, 3, 5, 5, 0, 9, 2, 7, 0, 1, 3, 3, 1, 4, 3, 3, 1, 6, 3, 8, 3, 8, 9, 6, 2, 9, 1, 8, 2, 9, 1, 6, 7, 1, 3, 0, 7, 6, 3, 2, 4, 0, 1, 6, 7, 3, 9, 6, 4, 6, 5, 3, 6, 8, 2, 7, 0, 9, 5, 6, 8, 2, 5, 1, 9, 3, 6, 2, 8, 8, 6, 7, 0, 6, 3, 2, 3, 5, 7, 3, 6, 2, 7, 8, 2, 1, 7, 7, 6, 8, 6, 5, 5, 1, 2, 8
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

3/Pi^2 is the limit of (Sum_{k=1..n} phi(k))/n^2, where phi(k) is Euler's totient A000010(k), i.e., of A002088(n)/A000290(n) as n tends to infinity.
The previous comment in the context of Farey series means that the length of the nth Farey series can be approximated by multiplying this constant by n^2, "and that the approximation gets proportionally better as n gets larger", according to Conway and Guy.  Alonso del Arte, May 28 2011
The density of the antitau numbers, A046642 (see Zelinsky link theorem 57 page 15).  Michel Marcus, May 31 2015
The asymptotic density of the sequences of squarefree numbers with even number of prime factors (A030229), odd number of prime factors (A030059), and coprime to 6 (A276378).  Amiram Eldar, May 22 2020


REFERENCES

J. H. Conway and R. K. Guy, The Book of Numbers, New York: SpringerVerlag, 1995, p. 156
L. E. Dickson, History of the Theory of Numbers, Vol. I pp. 126 Chelsea NY 1966.


LINKS

Table of n, a(n) for n=0..99.
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.


FORMULA

Equals Sum_{n>=1} 1/A039956(n)^2.  Amiram Eldar, May 22 2020
From Terry D. Grant, Oct 31 2020: (Start)
Equals (1)*zeta(0)/zeta(2).
Equals 1/(zeta(2)/2).
Equals 1/A195055.
Equals (1/2)*Sum_{k>=1} mu(k)/k^2.
(End)


EXAMPLE

3/Pi^2 = 0.303963550927013314331638389629...


MATHEMATICA

l = RealDigits[N[3/Pi^2, 100]]; Prepend[First[l], Last[l]] (* Ryan Propper, Aug 04 2005 *)


PROG

(PARI) 3/Pi^2 \\ Charles R Greathouse IV, Mar 08 2013


CROSSREFS

Cf. A000010, A002088, A000290.
Cf. A046642, A030229, A030059, A039956, A276378.
Cf. A013661, A195055, A306633, A082020, A088246.
Sequence in context: A132330 A117078 A021333 * A279977 A060533 A177785
Adjacent sequences: A104138 A104139 A104140 * A104142 A104143 A104144


KEYWORD

nonn,cons


AUTHOR

Lekraj Beedassy, Mar 07 2005


EXTENSIONS

More terms from Ryan Propper, Aug 04 2005


STATUS

approved



