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A082020 Decimal expansion of 15/Pi^2.

%I A082020 #64 Feb 16 2025 08:32:48
%S A082020 1,5,1,9,8,1,7,7,5,4,6,3,5,0,6,6,5,7,1,6,5,8,1,9,1,9,4,8,1,4,5,9,1,4,
%T A082020 5,8,3,5,6,5,3,8,1,6,2,0,0,8,3,6,9,8,2,3,2,6,8,4,1,3,5,4,7,8,4,1,2,5,
%U A082020 9,6,8,1,4,4,3,3,5,3,1,6,1,7,8,6,8,1,3,9,1,0,8,8,8,4,3,2,7,5,6
%N A082020 Decimal expansion of 15/Pi^2.
%C A082020 3/(2*Pi^2) (the same decimal expansion with an offset 0) is the probability that the greatest common divisor of two numbers selected at random is 2 (Christopher, 1956). - _Amiram Eldar_, May 23 2020
%C A082020 Sum of 1/n^2 over all squarefree n, see Penn link. - _Charles R Greathouse IV_, Jan 01 2022
%C A082020 Equals the asymptotic mean of the abundancy index of the cubefree numbers (A004709) (Jakimczuk and Lalín, 2022). - _Amiram Eldar_, May 12 2023
%H A082020 G. C. Greubel, <a href="/A082020/b082020.txt">Table of n, a(n) for n = 1..1000</a>
%H A082020 John Christopher, <a href="http://www.jstor.org/stable/2309400">The Asymptotic Density of Some k-Dimensional Sets</a>, The American Mathematical Monthly, Vol. 63, No. 6 (1956), pp. 399-401.
%H A082020 Werner Hürlimann, <a href="http://dx.doi.org/10.18642/jantaa_7100121599">Dedekind's arithmetic function and primitive four squares counting functions</a>, Journal of Algebra, Number Theory: Advances and Applications, Vol. 14, No. 2 (2015), pp. 73-88.
%H A082020 Rafael Jakimczuk and Matilde Lalín, <a href="https://doi.org/10.7546/nntdm.2022.28.4.617-634">Asymptotics of sums of divisor functions over sequences with restricted factorization structure</a>, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
%H A082020 Michael Penn, <a href="https://www.youtube.com/watch?v=1c5wZU347C8">Irregular numbers and the coolest sum I have ever seen!</a>, 2021 video.
%H A082020 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper4/page1.htm">Irregular numbers</a>, J. Indian Math. Soc., Vol. 5 (1913), pp. 105-106.
%H A082020 V. Sitaramaiah and M. V. Subbarao, <a href="https://doi.org/10.1007/BFb0086405">Some asymptotic formulae involving powers of arithmetic functions</a>, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, <a href="http://www.math.ualberta.ca/~subbarao/documents/Subbarao3.pdf">alternative link</a> (p. 230).
%H A082020 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>.
%H A082020 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusFunction.html">Moebius Function</a>.
%H A082020 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeProducts.html">Prime Products</a>.
%H A082020 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A082020 Product_{n >= 1} (1+1/prime(n)^2) = 15/Pi^2 (Ramanujan).
%F A082020 Equals zeta(2)/zeta(4) = A013661/A013662 = Sum_{n>=1} mu(n)^2/n^2 = Sum_{n>=1} |mu(n)|/n^2 . - _Enrique Pérez Herrero_, Jan 15 2012
%F A082020 Equals Sum_{n>=1} 1/A005117(n)^2 . - _Enrique Pérez Herrero_, Mar 30 2012
%F A082020 Equals lim_{n->oo} (1/n) * Sum_{k=1..n} psi(k)/k, where psi(k) is the Dedekind psi function (A001615). - _Amiram Eldar_, May 12 2019.
%F A082020 Equals Sum_{k>=1} A007434(k)/k^4. - _Amiram Eldar_, Jan 25 2024
%e A082020 1.51981775463506657...
%p A082020 evalf(15/Pi^2, 120); # _G. C. Greubel_, Oct 18 2019
%t A082020 A082020[digits_] := First[RealDigits[Zeta[2]/Zeta[4],10,digits]]; A082020[100] (* _Enrique Pérez Herrero_, Jan 15 2012 *)
%t A082020 RealDigits[15/Pi^2,10,120][[1]] (* _Harvey P. Dale_, Jun 23 2019 *)
%o A082020 (PARI) 15/Pi^2 \\ _Michel Marcus_, Oct 18 2019
%o A082020 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); 15/Pi(R)^2; // _G. C. Greubel_, Oct 18 2019
%o A082020 (Sage) numerical_approx(15/pi^2, digits=100) # _G. C. Greubel_, Oct 18 2019
%Y A082020 Cf. A001615, A004709, A005117, A007434, A013661, A013662, A157290.
%K A082020 nonn,cons
%O A082020 1,2
%A A082020 _N. J. A. Sloane_, May 09 2003