A393650 Decimal expansion of 9/Pi^2.

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

Offset

0,1

Comments

The asymptotic probability that the greatest common divisor of two positive integers selected independently at random is a 3-smooth number (A003586).

In general, the asymptotic probability that the greatest common divisor of two positive integers selected independently at random is a p-smooth number (number whose prime factors are all less than or equal to p) is Product_{primes q > p} (1 - 1/q^2) = (1/zeta(2)) / Product_{primes q <= p} (1 - 1/q^2).

The asymptotic probability that two 5-rough numbers (A007310) selected independently at random are coprime.

The asymptotic density of squarefree numbers within the 5-rough numbers (A276378 within A007310).

Links

Table of n, a(n) for n=0..105.

Formula

Equals 1/A100044.

Equals 3/(2*zeta(2)).

Equals (3/Pi)^2 = A089491^2.

Equals 3 * A104141.

Equals Sum_{n>=1} (-1)^(A005117(n)+1)/A005117(n)^2.

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} abs(A085097(k)).

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} abs(A092673(k)).

Equals lim_{m->oo} (1/(m*log(m))) * Sum_{k=1..m} A365208(k).

Equals lim_{m->oo} A104191(m)/4^m.

Equals lim_{m->oo} A179213(m)/m^2.

Example

0.911890652781039942994915168887548750139228972050218...

Mathematica

RealDigits[9/Pi^2, 10, 120][[1]]

Prog

(PARI) 9/Pi^2

Crossrefs

Cf. A003586, A005117, A007310, A013661, A065333, A085097, A089491, A092673, A100044, A104141, A104191, A132699, A179213, A276378, A365208.

The asymptotic probability that the greatest common divisor of two positive integers selected at random is: A010701 (not 5-rough), A010722 (5-rough), A020773 (even), A059956 (1), A082020/10 (2), A152627 (odd), A182448 (square), A185197 (even power of 2), A215267 (squarefree), A217739 (power of 2), A222056 (prime), A343359 (cubefree), A393646 (cube), A393647 (exponentially odd number), A393648 (powerful), A393649 (cubefull), this constant (3-smooth), A393651 (prime power), A393652 (perfect power).

Sequence in context: A388101 A388463 A092578 * A395848 A331247 A128060

Adjacent sequences: A393647 A393648 A393649 * A393651 A393652 A393653

Keyword

nonn,cons,easy

Author

Amiram Eldar, Feb 24 2026

Status

approved