A323669 - OEIS

A323669

Decimal expansion of 15/(2*Pi^2) = 1/((4/5)*zeta(2)).

7, 5, 9, 9, 0, 8, 8, 7, 7, 3, 1, 7, 5, 3, 3, 2, 8, 5, 8, 2, 9, 0, 9, 5, 9, 7, 4, 0, 7, 2, 9, 5, 7, 2, 9, 1, 7, 8, 2, 6, 9, 0, 8, 1, 0, 0, 4, 1, 8, 4, 9, 1, 1, 6, 3, 4, 2, 0, 6, 7, 7, 3, 9, 2, 0, 6, 2, 9, 8, 4, 0, 7, 2, 1, 6, 7, 6, 5

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OFFSET

0,1

COMMENTS

This is the limit n -> infinity of (1/n^2)*Phi_1(n) = (1/n^2)*Sum_{k=1..n} psi(k), with Dedekind's psi function psi(k) = k*Product_{p|k} (1 + 1/p) = A001615(k). Distinct primes p dividing k appear, and the empty product for k = 1 is set to 1. See the Walfisz reference, Satz 2., p. 100 (with x -> n, and phi_1(n) = psi(n)).

For the rationals r(n) = (1/n^2)*Phi_1(n) see A327340(n)/A327341(n), n >= 1.

From Amiram Eldar, Feb 24 2026: (Start)

(15/(2*Pi^2))/10 = 3/(4*Pi^2) is the asymptotic probability that the greatest common divisor of two positive integers selected independently at random is a power of 3 that is greater than 1.

In general, the asymptotic probability that the greatest common divisor of two positive integers selected independently at random is a power greater than 1 of a given prime p equals 1/(zeta(2)*(p^2-1)). If p = prime(k) and k >= 3, then this probability also equals 1/(A084922(k)*Pi^2). (End)

REFERENCES

Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 100, Satz 2.

LINKS

Table of n, a(n) for n=0..77.

Eric Weisstein's World of Mathematics, Dedekind Function.

Wikipedia, Dedekind psi function.

Index entries for transcendental numbers.

FORMULA

Equals 15/(2*Pi^2) = 1/((4/5)*zeta(2)), with 1/zeta(2) = A059956.

EXAMPLE

0.7599088773175332858290959740729572917826908100418491163420677392062984...

MATHEMATICA