cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A183699 Decimal expansion of zeta(2)*zeta(3), the product of two Riemann zeta values.

Original entry on oeis.org

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

Views

Author

R. J. Mathar, Jan 06 2011

Keywords

Comments

Equals the Dirichlet zeta-function Sum_{n>=1} A000203(n)/n^s at s=3.

Examples

			Equals 1.977304350297296118197085441485...

Crossrefs

Cf. A000203, A013661, A002117, A347213.
Cf. A183700 (s=4).

Programs

  • Maple
    evalf(Zeta(2)*Zeta(3));
  • Mathematica
    RealDigits[Zeta[2] * Zeta[3], 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)
  • PARI
    zeta(2)*zeta(3) \\ Charles R Greathouse IV, Mar 04 2015

Formula

Equals A013661 * A002117.

A288419 a(n) = Sum_{d|n} d^3*A000593(n/d).

Original entry on oeis.org

1, 9, 31, 73, 131, 279, 351, 585, 850, 1179, 1343, 2263, 2211, 3159, 4061, 4681, 4931, 7650, 6879, 9563, 10881, 12087, 12191, 18135, 16406, 19899, 22990, 25623, 24419, 36549, 29823, 37449, 41633, 44379, 45981, 62050, 50691, 61911, 68541, 76635, 68963, 97929
Offset: 1

Views

Author

Seiichi Manyama, Jun 09 2017

Keywords

Comments

Multiplicative because this sequence is the Dirichlet convolution of A000578 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

Links

Crossrefs

Cf. A000578, A027847, A183700, A288414.
Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), A288418 (k=2), this sequence (k=3), A288420 (k=4).

Programs

  • Mathematica
    f[p_, e_] := (p^(3*e+5) - (p^2+p+1)*p^(e+1) + p + 1)/((p^3-1)*(p^2-1)); f[2, e_] := (8^(e+1)-1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)
  • PARI
    a(n)={sumdiv(n, d, (n/d)^3*sigma(d>>valuation(d,2)))} \\ Andrew Howroyd, Jul 27 2018

Formula

From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A027847(n) for odd n.
Multiplicative with a(2^e) = (8^(e+1)-1)/7 and a(p^e) = (p^(3*e+5) - (p^2+p+1)*p^(e+1) + p + 1)/((p^3-1)*(p^2-1)) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^4, where c = 7*Pi^4*zeta(3)/2880 = (7/32)*zeta(3)*zeta(4) = (7/32) * A183700 = 0.284596... . (End)

A369720 The sum of divisors of the smallest cubefull number that is a multiple of n.

Original entry on oeis.org

1, 15, 40, 15, 156, 600, 400, 15, 40, 2340, 1464, 600, 2380, 6000, 6240, 31, 5220, 600, 7240, 2340, 16000, 21960, 12720, 600, 156, 35700, 40, 6000, 25260, 93600, 30784, 63, 58560, 78300, 62400, 600, 52060, 108600, 95200, 2340, 70644, 240000, 81400, 21960, 6240
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2024

Keywords

Links

Crossrefs

Cf. A000203, A036966, A356193, A369717, A369719, A369721.
Cf. A002117, A013662, A183700.

Programs

  • Mathematica
    f[p_, e_] := (p^If[e <= 2, 4, e + 1]-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
  • PARI
    a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i,2] <= 2, f[i,2] = 3)); sigma(f);}

Formula

a(n) = A000203(A356193(n)).
Multiplicative with a(p) = p^3 + p^2 + p + 1 for e <= 2, and a(p^e) = (p^(e+1)-1)/(p-1) for e >= 3.
a(n) >= A000203(n), with equality if and only if n is cubefull (A036966).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*s-3) - 1/p^(2*s-2) + 1/p^(4*s-4)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^3 - 1/p^4 + 1/p^7 + 1/p^12 - 1/p^13) = 1.00015013207437782094... .

A369721 The sum of unitary divisors of the smallest cubefull number that is a multiple of n.

Original entry on oeis.org

1, 9, 28, 9, 126, 252, 344, 9, 28, 1134, 1332, 252, 2198, 3096, 3528, 17, 4914, 252, 6860, 1134, 9632, 11988, 12168, 252, 126, 19782, 28, 3096, 24390, 31752, 29792, 33, 37296, 44226, 43344, 252, 50654, 61740, 61544, 1134, 68922, 86688, 79508, 11988, 3528, 109512
Offset: 1

Views

Author

Amiram Eldar, Jan 30 2024

Keywords

Links

Crossrefs

Cf. A034448, A036966, A356193, A369718, A369719, A369720.
Cf. A002117, A013662, A183700.

Programs

  • Mathematica
    f[p_, e_] := If[e <= 2, p^3 + 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= 2, 1 + f[i,1]^3, 1 + f[i,1]^f[i,2]));}

Formula

a(n) = A034448(A356193(n)).
Multiplicative with a(p) = p^3 + 1 for e <= 2, and a(p^e) = p^e + 1 for e >= 3.
a(n) >= A034448(n), with equality if and only if n is cubefull (A036966).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-3) - 1/p^(s-1) - 1/p^(2*s-4) + 1/p^(4*s-4) - 1/p^(4*s-3) ).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^12 - 2/p^13 + 1/p^14) = 0.65803546696642353777... .

A123733 Decimal expansion of Sum_{m>=1} (-1)Sigma(m)/m^4.

Original entry on oeis.org

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

Views

Author

Yasutoshi Kohmoto, Nov 18 2006

Keywords

Examples

			1.12634286431455070572497848729992683258464206350262...

Crossrefs

Cf. A049060, A123732, A123734, A183700.

Programs

  • PARI
    prodeulerrat(1 + (p^4-p^3+1)/((p-1)^2*(p^2+p+1)*(p^3+p^2+p+1))) \\ Amiram Eldar, Aug 26 2022

Formula

Equals Sum_{m>=1} A049060(m)/m^4.
Equals Product_{p prime} (1 + (p^4- p^3+1)/((p-1)^2*(p^2+p+1)*(p^3+p^2+p+1))). - Amiram Eldar, Aug 26 2022

Extensions

46 more digits from R. J. Mathar, Dec 19 2010
More terms from Amiram Eldar, Aug 26 2022

A383289 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^2 dx dy dz, where {w} is the fractional part of w.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 26 2025

Keywords

Examples

			0.02340961823158087268020093855006980675840442582714...

Links

Crossrefs

Cf. A002117, A013661, A013662, A013664, A183699, A183700.
Cf. A375901 (m=1), this constant (m=2), A386564 (m=3).

Programs

  • Mathematica
    RealDigits[1 - Zeta[2]/2 - Zeta[3]/2 + 7*Zeta[6]/48 + Zeta[2]*Zeta[3]/18 + Zeta[3]^2/18 + Zeta[3]*Zeta[4]/12, 10, 120, -1][[1]]
RealDigits[With[{m = 2}, 1 - 3*Sum[Zeta[j + 1], {j, 1, m}]/(2*(m + 1)) + Sum[Zeta[j + 1], {j, 1, m}] * Sum[(j + 1)*Zeta[j + 2], {j, 1, m}]/((m + 1)^2*(m + 2))], 10, 106][[1]] (* Vaclav Kotesovec, Jul 26 2025, following the general formula found by the solvers *)
  • PARI
    1 - zeta(2)/2 - zeta(3)/2 + 7*zeta(6)/48 + zeta(2)*zeta(3)/18 + zeta(3)^2/18 + zeta(3)*zeta(4)/12

Formula

Equals 1 - zeta(2)/2 - zeta(3)/2 + 7*zeta(6)/48 + zeta(2)*zeta(3)/18 + zeta(3)^2/18 + zeta(3)*zeta(4)/12.
In general, Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^m dx dy dz = 1 - 3*Sum_{j=1..m} zeta(j+1)/(2*(m+1)) + (Sum_{j=1..m} zeta(j+1))*(Sum_{j=1..m} (j+1)*zeta(j+2))/((m+1)^2*(m+2)).

A386564 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^3 dx dy dz, where {w} is the fractional part of w.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 26 2025

Keywords

Examples

			0.00778895508409665205428360965992714119017196489266...

Links

  • Cornel Ioan Vălean, Problem 11902, Problems and Solutions, The American Mathematical Monthly, Vol. 123, No. 4 (2016), p. 399; A Row of Zetas, Solution to Problem 11902 by Rituraj Nandan, ibid., Vol. 125, No. 2 (2018), pp. 182-184.
  • Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer (2019), section 1.48 The Calculation of a Beautiful Triple Fractional Part Integral with a Cubic Power, p. 31.

Crossrefs

Cf. A002117, A013661, A013662, A013663, A013664, A013666, A183699, A183700.
Cf. A375901 (m=1), A383289 (m=2), this constant (m=3).

Programs

  • Mathematica
    RealDigits[1 - 3*(Zeta[2]+Zeta[3]+Zeta[4])/8 + 21*Zeta[6]/320 + 7*Zeta[8]/160 + Zeta[3]^2/40 + Zeta[2]*Zeta[3]/40 + Zeta[2]*Zeta[5]/20 + Zeta[3]*Zeta[4]/16 + Zeta[3]*Zeta[5]/20 + Zeta[4]*Zeta[5]/20, 10, 120, -1][[1]]
  • PARI
    1 - 3*(zeta(2)+zeta(3)+zeta(4))/8 + 21*zeta(6)/320 + 7*zeta(8)/160 + zeta(3)^2/40 + zeta(2)*zeta(3)/40 + zeta(2)*zeta(5)/20 + zeta(3)*zeta(4)/16 + zeta(3)*zeta(5)/20 + zeta(4)*zeta(5)/20

Formula

Equal 1 - 3*(zeta(2)+zeta(3)+zeta(4))/8 + 21*zeta(6)/320 + 7*zeta(8)/160 + zeta(3)^2/40 + zeta(2)*zeta(3)/40 + zeta(2)*zeta(5)/20 + zeta(3)*zeta(4)/16 + zeta(3)*zeta(5)/20 + zeta(4)*zeta(5)/20.
In general, Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^m dx dy dz = 1 - 3*Sum_{j=1..m} zeta(j+1)/(2*(m+1)) + (Sum_{j=1..m} zeta(j+1))*(Sum_{j=1..m} (j+1)*zeta(j+2))/((m+1)^2*(m+2)).
Showing 1-7 of 7 results.

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