A104141 Decimal expansion of 3/Pi^2.

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

Offset: 0

Author

Lekraj Beedassy, Mar 07 2005

Keywords

• nonn
• cons

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 n-th 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 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

Examples

			3/Pi^2 = 0.303963550927013314331638389629...
		

References

• J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, p. 156.
• L. E. Dickson, History of the Theory of Numbers, Vol. I pp. 126 Chelsea NY 1966.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 184.

Links

• Eric Weisstein's World of Mathematics, Farey Sequence
• Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000010, A002088, A000290.

Cf. A046642, A030229, A030059, A039956, A276378.

Cf. A013661, A195055, A306633, A082020, A088246.

Programs

Mathematica

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

PARI

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

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)

From Hugo Pfoertner, Apr 23 2024: (Start)

Equals A059956/2.

Equals A082020/5. (End)

Extensions

More terms from Ryan Propper, Aug 04 2005