A233091 -id:A233091 - cp's OEIS frontend

A377558 Decimal expansion of Pi^3/64 + 7*zeta(3)/16.

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

Offset: 1

Stefano Spezia, Nov 01 2024

			1.01037296826200719010420286858471867099445163674...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 40.

Cf. A002117, A016813, A016815, A091925, A233091, A377557, A377559, A377560.

RealDigits[Pi^3/64+7Zeta[3]/16,10,100][[1]]

Equals Sum_{k>=0} 1/(4*k + 1)^3 (see Finch).
Equals -psi''(1/4)/128 = -(psi''(1/8) + psi''(5/8))/1024 (see Shamos).
Equals hypergeom([1/4, 1/4, 1/4, 1], [5/4, 5/4, 5/4], 1). - R. J. Mathar, Jul 14 2025

A377559 Decimal expansion of Sum_{k>=0} 1/(5*k + 1)^3.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 01 2024

			1.00591214445774373223679236014700144825493611206...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 32.

Cf. A016861, A016863, A233091, A377557, A377558, A377560.

RealDigits[-PolyGamma[2,1/5]/250,10,100][[1]]

Equals -psi''(1/5)/250 (see Shamos).

A377560 Decimal expansion of Pi^3/(36sqrt(3)) + 91zeta(3)/216.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 01 2024

			1.00368551534795269706323013702486057315272784359...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 28.

Cf. A002117, A002194, A016921, A016923, A091925, A233091, A377557, A377558, A377559.

RealDigits[Pi^3/(36*Sqrt[3])+91*Zeta[3]/216,10,100][[1]]

Equals Sum_{k>=0} 1/(6*k + 1)^3 (see Finch).

Equals -psi''(1/6)/432 (see Shamos).

A381822 Odd cubefree numbers: odd numbers that are not divisible by any cube greater than 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 127, 129, 131

Offset: 1

Amiram Eldar, Mar 08 2025

Numbers whose prime factorization has only odd primes, and all its exponents are smaller than 3 (except for 1 whose prime factorization is empty).
The asymptotic density of this sequence is 4/(7*zeta(3)) = 1/(2*A233091) = 0.475375641474689982104... .
In general, the asymptotic density of odd k-free numbers (numbers that are not divisible by a k-th power other than 1, k >= 2) is 2^(k-1)/((2^k-1) * zeta(k)).

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

Intersection of A005408 and A004709.
A056911 is a subsequence.
Cf. A002117, A233091.

cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;;, 2]], # < 3 &]; Select[Range[1, 150, 2], cubeFreeQ]

isok(k) = k % 2 && if(k == 1, 1, vecmax(factor(k)[, 2]) < 3);

A273839 Decimal expansion the Bessel moment c(4,0) = Integral_{0..inf} K_0(x)^4 dx, where K_0 is the modified Bessel function of the second kind.

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, Jun 01 2016

			27.2413384178059734067099802645579350239978880986182746551229...

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891.

Cf. A273816 (c(3,0)), A273817 (c(3,1)), A273818 (c(3,2)), A273819 (c(3,3)), A233091 (c(4,1)), A273840 (c(4,2)), A273841 (c(4,3)).

c[4, 0] = (Pi^4/4)*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {1, 1, 1}, 1];
RealDigits[c[4, 0], 10, 102][[1]]

c(4,0) = (Pi^4/4) Sum_{n>=0} binomial(2n, n)^4/2^(8n).

Equals (Pi^4/4) 4F3(1/2, 1/2, 1/2, 1/2; 1, 1, 1; 1), where 4F3 is the generalized hypergeometric function.

A273840 Decimal expansion the Bessel moment c(4,2) = Integral_{0..inf} x^2 K_0(x)^4 dx, where K_0 is the modified Bessel function of the second kind.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 01 2016

			0.195770625247287917217458083275577237418827896966525028197933846...

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891.

Cf. A273816 (c(3,0)), A273817 (c(3,1)), A273818 (c(3,2)), A273819 (c(3,3)), A273839 (c(4,0)), A233091 (c(4,1)), A273841 (c(4,3)).

c[4, 2] = (Pi^4/64)*(4*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {1, 1, 1}, 1] - 3*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {2, 1, 1}, 1]) - 3*Pi^2 / 16;
RealDigits[c[4, 2], 10, 104][[1]]

c(4,2) = (Pi^4/64)*(4 * 4F3(1/2, 1/2, 1/2, 1/2; 1, 1, 1; 1) - 3 * 4F3(1/2, 1/2, 1/2, 1/2; 2, 1, 1; 1)) - 3*Pi^2/16, where 4F3 is the generalized hypergeometric function.

A273841 Decimal expansion the Bessel moment c(4,3) = Integral_{0..inf} x^3 K_0(x)^4 dx, where K_0 is the modified Bessel function of the second kind.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 01 2016

			0.075449947566161249931192722830629685479840751449484130392059402731...

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891.
R. J. Mathar, Some definite intgrals over a power multiplied by four modified Bessel functions vixra:1606.0141 (2016) eq. (64)

Cf. A273816 (c(3,0)), A273817 (c(3,1)), A273818 (c(3,2)), A273819 (c(3,3)), A273839 (c(4,0)), A233091 (c(4,1)), A273840 (c(4,2)).

c[4, 3] = (7/32)*Zeta[3] - 3/16;
RealDigits[c[4, 3], 10, 103][[1]]

zeta(3)*7/32-3/16 \\ Charles R Greathouse IV, Oct 23 2023

c(4,3) = (7/32)*zeta(3) - 3/16.

A247450 Decimal expansion of c(4), a constant appearing in certain Euler double sums not expressible in terms of well-known constants.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 17 2014

			2.117141734777039411129100226012451751917680766916084...

J. M. Borwein, I.J. Zucker and J. Boersma, The evaluation of character Euler double sums, The Ramanujan Journal, April 2008, Volume 15, Issue 3, pp 377-405, see p. 17 c(4).

Cf. A002162 c(1), A072691 c(2), A233091 c(3).

c[4] = (1/12)*((-Pi^2)*Log[2]^2 + Log[2]^4 + 24*PolyLog[4, 1/2] + 21*Log[2]*Zeta[3]); RealDigits[c[4], 10, 104] // First

c(n) = sum_{k=0..n-2} (n-2)!/k!*log(2)^k*Li_(n-k)(1/2) + log(2)^n/n.

c(4) = (1/12)*((-Pi^2)*log(2)^2 + log(2)^4 + 24*Li_4(1/2) + 21*log(2)*zeta(3)).

A251967 Decimal expansion of 29*Pi^3/864.

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Dec 12 2014

			1.040719934871174519778718908502245903778395102327162179548832876169...

Bruno Berselli, Sums of the type sum(i>=0, (-1)^k/(2i+1)^3).
Index entries for transcendental numbers

Cf. similar sums:
A233091 for Sum_{i>=0} 1/(2i+1)^3;
A153071 for Sum_{i>=0} (-1)^i/(2i+1)^3;
A251809 for Sum_{i>=0} (-1)^floor(i/2)/(2i+1)^3.

Mathematica
```
RealDigits[29 Pi^3/864, 10, 90][[1]]
```

29*Pi^3/864 \\ Charles R Greathouse IV, Oct 01 2022

Equals Sum_{i>=0} (-1)^floor(i/3)/(2i+1)^3 = +1 +1/3^3 +1/5^3 -1/7^3 -1/9^3 -1/11^3 +1/13^3 +1/15^3 +1/17^3 - ...

A381313 Numbers that are divisible by the cube of an odd prime.

Original entry on oeis.org

27, 54, 81, 108, 125, 135, 162, 189, 216, 243, 250, 270, 297, 324, 343, 351, 375, 378, 405, 432, 459, 486, 500, 513, 540, 567, 594, 621, 625, 648, 675, 686, 702, 729, 750, 756, 783, 810, 837, 864, 875, 891, 918, 945, 972, 999, 1000, 1026, 1029, 1053, 1080, 1107, 1125

Offset: 1

Amiram Eldar, Feb 19 2025

Numbers whose odd part is not cubefree.

The asymptotic density of this sequence is 1 - 8/(7*zeta(3)) = 1 - 1/A233091 = 0.04924871705062003579... .

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

Subsequence of A046099 and A038838.

Cf. A000265, A002117, A233091.

cubeFreeQ[k_] := Max[FactorInteger[k][[;;, 2]]] < 3; q[k_] := !cubeFreeQ[k / 2^IntegerExponent[k, 2]]; Select[Range[1200], q]

iscubefree(k) = if(k == 1, 1, vecmax(factor(k)[, 2]) < 3);
isok(k) = !iscubefree(k >> valuation(k, 2));

cp's OEIS Frontend

A377558 Decimal expansion of Pi^3/64 + 7*zeta(3)/16.

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A377559 Decimal expansion of Sum_{k>=0} 1/(5*k + 1)^3.

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A377560 Decimal expansion of Pi^3/(36*sqrt(3)) + 91*zeta(3)/216.

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A381822 Odd cubefree numbers: odd numbers that are not divisible by any cube greater than 1.

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A273839 Decimal expansion the Bessel moment c(4,0) = Integral_{0..inf} K_0(x)^4 dx, where K_0 is the modified Bessel function of the second kind.

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A273840 Decimal expansion the Bessel moment c(4,2) = Integral_{0..inf} x^2 K_0(x)^4 dx, where K_0 is the modified Bessel function of the second kind.

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A273841 Decimal expansion the Bessel moment c(4,3) = Integral_{0..inf} x^3 K_0(x)^4 dx, where K_0 is the modified Bessel function of the second kind.

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A247450 Decimal expansion of c(4), a constant appearing in certain Euler double sums not expressible in terms of well-known constants.

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A377560 Decimal expansion of Pi^3/(36sqrt(3)) + 91zeta(3)/216.