A088246 -id:A088246 - cp's OEIS frontend

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

Lekraj Beedassy, Mar 07 2005

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

			3/Pi^2 = 0.303963550927013314331638389629...

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.

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.

Cf. A000010, A002088, A000290.
Cf. A046642, A030229, A030059, A039956, A276378.
Cf. A013661, A195055, A306633, A082020, A088246.

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

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

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)

More terms from Ryan Propper, Aug 04 2005

A088245 Decimal expansion of 9/(2*Pi^2).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Sep 25 2003

The asymptotic density of squarefree numbers not divisible by 3 (A261034). - Amiram Eldar, May 22 2020

			0.455945326390519971497457584443774375...

See the Hardy reference given under A030059, eq. (4.9.4), p. 64, from the corrected formula on p. 65 for s=2. - Wolfdieter Lang, Oct 18 2016

Eric Weisstein's World of Mathematics, Moebius Function
Index entries for transcendental numbers

Cf. A082020, A088246, A030059, A261034.

RealDigits[N[9/(2 Pi^2), 120]] // First (* Michael De Vlieger, Oct 23 2015 *)

PARI
```
9/(2*Pi^2) \\ Altug Alkan, Oct 23 2015
```

Equals Sum_{n > 0} 1/A030059(n)^2 (the sum of reciprocals of squarefree numbers with an odd number of prime factors). Convergence is very slow. - Michel Lagneau, Oct 23 2015

A372962 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^2.

Original entry on oeis.org

1, 29, 235, 925, 3101, 6815, 16759, 29597, 57097, 89929, 160931, 217375, 371125, 486011, 728735, 947101, 1419569, 1655813, 2475739, 2868425, 3938365, 4666999, 6435815, 6955295, 9690601, 10762625, 13874563, 15502075, 20510309, 21133315, 28628191, 30307229, 37818785

Offset: 1

Seiichi Manyama, May 18 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A068963, A084218, A372963.
Cf. A001160, A008683.
Cf. A013662, A013664, A088246.
Cf. A350156, A372964.
Cf. A371492, A372952.

f[p_, e_] := (p^(5*e+5) - p^(5*e+2) + p^2 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^2*sigma(d, 5));

a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_5(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+2) + p^2 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(4) = 2*Pi^2/21 = 0.939962323... (1/A088246). (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^4 * sigma_4(d^2)/sigma_2(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^3. - Seiichi Manyama, May 25 2024

A372927 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5.

Original entry on oeis.org

1, 35, 251, 1132, 3149, 8785, 16855, 36272, 61065, 110215, 161171, 284132, 371461, 589925, 790399, 1160896, 1420145, 2137275, 2476459, 3564668, 4230605, 5640985, 6436871, 9104272, 9841225, 13001135, 14839443, 19079860, 20511989, 27663965, 28630111, 37149440, 40453921

Offset: 1

Seiichi Manyama, May 17 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A069097, A360428, A368743, A372926.
Cf. A001158, A008683.
Cf. A013662, A013664, A088246.

f[p_, e_] := p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^2*sigma(d, 3));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^2.
a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_3(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-5)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(4)/zeta(6) = 21/(2*Pi^2) = 1.0638724... (A088246). (End)

A383647 Decimal expansion of 15/(2*Pi^4).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, May 03 2025

			0.07699486691013251391864587450339020606370851390...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 64.

Jakub Buczak, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A030059, A082020, A088245, A088246, A092425, A151927.

Join[{0},RealDigits[15/(2Pi^4),10,100][[1]]]

Equals Sum_{n > 0} 1/A030059(n)^4.

Equals 10/A151927. - Hugo Pfoertner, May 03 2025

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A104141 Decimal expansion of 3/Pi^2.

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A088245 Decimal expansion of 9/(2*Pi^2).

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A372962 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^2.

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A372927 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5.

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A383647 Decimal expansion of 15/(2*Pi^4).

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