A100044 -id:A100044 - cp's OEIS frontend

A195056 Decimal expansion of Pi^2/7.

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

Offset: 1

Omar E. Pol, Oct 04 2011

			1.409943485869908374119212999982307305045...

F. Aubonnet, D. Guinin and B.Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A000265, A002388, A013661, A019674, A100044, A111003, A164102.

Pi(RealField(128))^2/7; // G. C. Greubel, Jun 02 2021

RealDigits[Pi^2/7, 10, 105][[1]] (* T. D. Noe, Oct 05 2011 *)

PARI
```
Pi^2/7 \\ Michel Marcus, Feb 04 2022
```

numerical_approx(pi^2/7, digits=128) # G. C. Greubel, Jun 02 2021

Equals Sum_{k>=1} A000265(k)/k^3. - Amiram Eldar, Jun 27 2020

Equals Integral_{x=0..1} log(1+x+x^2+x^3+x^4+x^5+x^6)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022

Extended by T. D. Noe, Oct 05 2011

A195057 Decimal expansion of Pi^2/11.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Oct 04 2011

			0.8972367637353962380758628181705591941194...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for transcendental numbers

Cf. A002388, A013661, A019678, A072691, A100044, A195059.

Pi(RealField(129))^2/11; // G. C. Greubel, Jun 03 2021

RealDigits[Pi^2/11, 10, 105][[1]] (* T. D. Noe, Oct 05 2011 *)

numerical_approx(pi^2/11, digits=128) # G. C. Greubel, Jun 03 2021

Extended by T. D. Noe, Oct 05 2011

A366442 The sum of divisors of the 5-rough numbers (A007310).

Original entry on oeis.org

1, 6, 8, 12, 14, 18, 20, 24, 31, 30, 32, 48, 38, 42, 44, 48, 57, 54, 72, 60, 62, 84, 68, 72, 74, 96, 80, 84, 108, 90, 112, 120, 98, 102, 104, 108, 110, 114, 144, 144, 133, 156, 128, 132, 160, 138, 140, 168, 180, 150, 152, 192, 158, 192, 164, 168, 183, 174, 248

Offset: 1

Amiram Eldar, Oct 10 2023

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

Cf. A000203, A013661, A007310, A100044, A366441.

Similar sequences: A008438, A062822, A065764, A180114, A362986, A366439, A366440.

a[n_] := DivisorSigma[1, 2*Floor[3*n/2] - 1]; Array[a, 100]

PARI
```
a(n) = sigma((3*n)\2 << 1 - 1)
```

from sympy import divisor_sigma
def A366442(n): return divisor_sigma((n+(n>>1)<<1)-1) # Chai Wah Wu, Oct 10 2023

a(n) = A000203(A007310(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2) = 1.644934... (A013661).
The asymptotic mean of the abundancy index of the 5-rough numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A007310(k) = Pi^2/9 = 1.0966227... (A100044).
In general, the asymptotic mean of the abundancy index of the prime(k)-rough numbers is zeta(2) * Product_{i=1..k-1} (1 - 1/prime(i)^2).

A282496 'Somos expansion' of Pi: Pi=a(0)sqrt(a(1)sqrt(a(2)sqrt(a(3)sqrt(...)))). a(n)=floor(x(n)), x(n)=x(n-1)^2/a(n-1)^2, x(0)=Pi.

Original entry on oeis.org

3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1

Offset: 0

Yuriy Sibirmovsky, Feb 16 2017

nonn

1<=a(n)<=3 for all n. Reasoning: for x>1 it follows that 1

			Integer part of Pi is 3. Integer part of Pi^2/9 is 1.

Yuriy Sibirmovsky, Table of n, a(n) for n = 0..1999

Cf. A000796 (digits), A100044 (Pi^2/9), A001203 (continued fraction), A276459 (another nested radical expansion).

$MaxExtraPrecision = 1000;
x00 = Pi;
x0 = x00;
Nm = 130;
j = 1;
Res = Table[1, {j, 1, Nm}];
While[j < Nm, Res[[j]] = Floor[x0]; x0 = N[(x0/ Res[[j]])^2, 20000];
  j++];
Res

Product_{k>=0} a(k)^(1/2^k) = Pi.

A373506 Decimal expansion of 4*Pi/3^(3/2) - Pi^2/9.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 07 2024

			1.321776441080139509810494232425524183566...

Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27.

Cf. A100044, A275486, A073016 (no n+1 denominator), A073010 (denominator n), A373507 (denominator n-1).

Maple
```
4*Pi/3^(3/2)-Pi^2/9 ; evalf(%) ;
```

RealDigits[4*Pi/3^(3/2) - Pi^2/9, 10, 120][[1]] (* Amiram Eldar, Jun 10 2024 *)

4*Pi/3^(3/2) - Pi^2/9 \\ Amiram Eldar, Jun 10 2024

Equals Sum_{n>=0} 1/((n+1)*binomial(2n,n)).
The alternating case is Sum_{n>=0} (-1)^n/((n+1)*binomial(2*n,n)) = 8*log(phi)/sqrt(5)-4*log^2(phi) = 0.79537... where phi is the golden ratio.
Equals A275486 - A100044. - Stefano Spezia, Jun 07 2024

A373508 Decimal expansion of sqrt(3)*Pi/6 + Pi^2/36 - 1.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jun 07 2024

			0.181055359925146664709108323262082...

Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27.

Cf. A093766, A100044, A373507 (denominator n-1), A073010 (denominator n).

Maple
```
(sqrt(3)+Pi/6)*Pi/6-1; evalf(%) ;
```

RealDigits[Sqrt[3]*Pi/6 + Pi^2/36 - 1, 10, 120][[1]] (* Amiram Eldar, Jun 10 2024 *)

sqrt(3)*Pi/6 + Pi^2/36 - 1 \\ Amiram Eldar, Jun 10 2024

Equals Sum_{n>=2} 1/((n-1)^2*binomial(2n,n)).
Sum_{n>=2} (-1)^n/((n-1)^2*binomial(2n,n)) = 1 - sqrt(5)*log(phi) + log(phi)^2 = 0.1555... where phi is the golden ratio.
Equals A093766 + A100044/4 - 1. - Stefano Spezia, Jun 07 2024

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A195056 Decimal expansion of Pi^2/7.

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A195057 Decimal expansion of Pi^2/11.

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A366442 The sum of divisors of the 5-rough numbers (A007310).

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A282496 'Somos expansion' of Pi: Pi=a(0)*sqrt(a(1)*sqrt(a(2)*sqrt(a(3)*sqrt(...)))). a(n)=floor(x(n)), x(n)=x(n-1)^2/a(n-1)^2, x(0)=Pi.

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A373506 Decimal expansion of 4*Pi/3^(3/2) - Pi^2/9.

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A373508 Decimal expansion of sqrt(3)*Pi/6 + Pi^2/36 - 1.

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A282496 'Somos expansion' of Pi: Pi=a(0)sqrt(a(1)sqrt(a(2)sqrt(a(3)sqrt(...)))). a(n)=floor(x(n)), x(n)=x(n-1)^2/a(n-1)^2, x(0)=Pi.