A021002 -id:A021002 - cp's OEIS frontend

A244285 Decimal expansion of A1B1, the average number of non-isomorphic semisimple rings of any order, where A1 is Product_{m>1} zeta(m) and B1 is Product_{rm^2 > 1} zeta(r*m^2).

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

Offset: 1

Jean-François Alcover, Jun 25 2014

The asymptotic mean of A038538. - Amiram Eldar, Jan 31 2024

			2.499616112936298274932373821756498034570402258807595443206210948121224368...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, pp. 274-276.

Cf. A021002 (A1), A038538, A123030.

Digits := 200: z:=product(Zeta(1.0*j), j = 2..1000): for k from 10 by 10 to 50 do print(z*product(product(Zeta(1.0*r*m^2), r = 1..k^2), m = 2..k)); end do; # Vaclav Kotesovec, Jun 11 2020

digits = 20; digitsPlus = 100; n0 = 50; dn = 1; A1 = NProduct[Zeta[m], {m, 2, Infinity}, WorkingPrecision -> digitsPlus]; Clear[B1]; B1[n_] := B1[n] = NProduct[Zeta[r*m^2], {r, 1, n}, {m, 2, n}, WorkingPrecision -> digitsPlus]; B1[n0]; B1[n = n0 + dn]; While[ RealDigits[B1[n], 10, digitsPlus] != RealDigits[B1[n - dn], 10, digitsPlus], Print["n = ", n]; n = n + dn]; RealDigits[A1*B1[n], 10, digits] // First

prodinf(m = 2, zeta(m)) * prodinf(r = 1, prodinf(m = 2, zeta(r*m^2))) \\ Amiram Eldar, Jan 31 2024

More digits from Vaclav Kotesovec, Jun 11 2020

A082868 Continued fraction approximation to Product_{k>=2} zeta(k).

Original entry on oeis.org

2, 3, 2, 1, 1, 4, 10, 1, 1, 2, 5, 1, 1, 3, 10, 3, 2, 11, 1, 2, 3, 6, 2, 1, 3, 9, 2, 1, 2, 7, 1, 1, 2, 26, 1, 21, 13, 1, 1, 1, 42, 2, 1, 2, 1, 4, 2, 1, 9, 18, 2, 2, 1, 1, 45, 2, 1, 1, 36, 4, 1, 5, 14, 2, 6, 2, 4, 1, 5, 3, 2, 3, 1, 4, 1, 2, 2, 13, 1, 3, 48, 23, 1, 1, 1, 7, 6, 1, 8, 1, 1, 4, 3, 3, 5, 2, 7, 2

Offset: 0

Wouter Meeussen, May 24 2003

Continued fraction of the constant A021002. - R. J. Mathar, Aug 05 2014

Cf. A021002.

ContinuedFraction@NProduct[Zeta[n], {n, 2, Infinity}, WorkingPrecision->200, AccuracyGoal->200, Method->SequenceLimit, NProductFactors->200, NProductExtraFactors->200]

A169862 Decimal expansion of root of x^(x-1) = (x-1)^x.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 18 2010, based on a suggestion from Daniel Forgues

			3.2931662874118610315080282912508058643722572903271212485377103961...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Foias Constant

Equals A085846 + 1. Cf. A021002.

RealDigits[x/.FindRoot[x^(x-1)==(x-1)^x,{x,3}, WorkingPrecision->150]] [[1]] (* Harvey P. Dale, Nov 09 2011 *)

solve(x=3,4,x^(x-1)-(x-1)^x) \\ Charles R Greathouse IV, Apr 14 2014

A369634 Decimal expansion of the infinite product of the Zeta Functions with arguments that are multiples of 3.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 28 2024

Dirichlet generating function of A000688 evaluated at s=3.

			1.22570470512849740952045767158897448248993384223224...

Cf. A000688, A021002, A080729.

Cf. A002117, A013664, A013667, A013670.

evalf(product(Zeta(3*k), k = 1 .. infinity), 120) # Amiram Eldar, Jan 28 2024

prodinf(k=1,zeta(3*k)) \\ Amiram Eldar, Jan 28 2024

Equals Product_{k>=1} zeta(3*k) = A002117 * A013664 * A013667 * A013670 *...

A249141 Decimal expansion of 'sigma', a constant associated with the expected number of random elements to generate a finite abelian group.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 22 2014

			2.11845656347016353238252776910236476428859...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 273.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 33.
Carl Pomerance, The expected number of random elements to generate a finite abelian group, Periodica Mathematica Hungarica 43 (2001), 191-198.

Cf. A021002, A033150, A084892, A084893.

digits = 102; jmax = 400; P[j_] := 1/Product[N[Zeta[k], digits+100], {k, j, jmax}]; sigma = 1+Sum[1 - P[j], {j, 2, jmax}]; RealDigits[sigma, 10, digits] // First

default(realprecision,120); 1 + suminf(j=2, 1 - prodinf(k=j, 1/zeta(k))) \\ Michel Marcus, Oct 22 2014

sigma = 1+sum_{j >= 2} (1-prod_{k >= j} zeta(k)^(-1)).

A328705 Dirichlet g.f.: Product_{k>=1} zeta(k*s)^2.

Original entry on oeis.org

1, 2, 2, 5, 2, 4, 2, 10, 5, 4, 2, 10, 2, 4, 4, 20, 2, 10, 2, 10, 4, 4, 2, 20, 5, 4, 10, 10, 2, 8, 2, 36, 4, 4, 4, 25, 2, 4, 4, 20, 2, 8, 2, 10, 10, 4, 2, 40, 5, 10, 4, 10, 2, 20, 4, 20, 4, 4, 2, 20, 2, 4, 10, 65, 4, 8, 2, 10, 4, 8, 2, 50, 2, 4, 10, 10, 4, 8, 2, 40

Offset: 1

Ilya Gutkovskiy, Oct 26 2019

Dirichlet convolution of A000688 with itself.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000688, A000712, A001620, A063966, A129667, A188581, A188585, A301830.

Table[DivisorSum[n, FiniteAbelianGroupCount[n/#] FiniteAbelianGroupCount[#] &], {n, 1, 80}]

a(n) = Sum_{d|n} A000688(n/d) * A000688(d).
Sum_{k=1..n} a(k) ~ c^2 * n * (log(n) + 2*gamma - 1 - 2*s), where c = A021002 = Product_{k>=2} zeta(k) = 2.2948565916733137941835158313443112887131637994..., s = Sum_{k>=2} k*zeta'(k)/zeta(k) = -2.1955691982567064617939038695473479681910375... and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 26 2019
Multiplicative with a(p^e) = A000712(e). - Amiram Eldar, Nov 30 2020

A335494 Decimal expansion of s, where s is the root of Product_{k>=1} zeta(k*s) = 2.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 11 2020

			1.8868691498777028184721160405750829854817368935543730821633756671766...

Eric Weisstein's World of Mathematics, Riemann Zeta Function.

Cf. A021002, A107311, A244285, A329385.

A338851 Constant T such that Sum_{n>=1} zeta(T*n)=1.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Nov 12 2020

			1.82024771787000913066...

Cf. A021002, A338815.

(*convergence test*)
Do[Print[FindRoot[Sum[Zeta[c n] - 1, {n, 1, m}] == 1, {c, 2},
   WorkingPrecision -> 200]], {m, 50, 450, 50}] (* Krzysztof Maslanka *)

A375887 Decimal expansion of Product_{n>=2} zeta(n)^n.

Original entry on oeis.org

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

Offset: 1

Richard R. Forberg, Sep 01 2024

It is interesting to note that this product is very close in value to 3 * Sum_{n>=2} (zeta(n)^n-1), A375920, where that factor's first 30 digits are: 3.00012312615292744064909403341.

			9.766825821453289699230682695640792162028987950967280928488833051400227...

Cf. A375920,(Sum_{n>=2} (zeta(n)^n-1)), A021002 (Product_{n>=2} zeta(n)), A093720 (Sum_{n >= 2} zeta(n)/n!), A013661 (zeta(2)).

evalf(Product(Zeta(n)^n, n = 2 .. infinity), 150); # Vaclav Kotesovec, Sep 02 2024

RealDigits[N[Product[Zeta[n]^n, {n, 2, 500}], 150]][[1]]

prodinf(k = 2, zeta(k)^k) \\ Amiram Eldar, Sep 02 2024

A375920 Decimal expansion of Sum_{n>=2} (zeta(n)^n - 1).

Original entry on oeis.org

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

Offset: 1

Richard R. Forberg, Sep 02 2024

It is interesting to note that this sum is very close in value to 1/3 of Product_{n>=2} zeta(n)^n, A375887, where that factor's first 30 digits are: 0.333319653211135001436063576617.

			3.255474995780369262094368665069015138075282643803397585341859272265720258...

Cf. A375887 (Product_{n>=2} zeta(n)^n), A021002 (Product_{n>2} zeta(n)), A093720 (Sum_{n>=2} zeta(n)/n!), A013661 (zeta(2)).

evalf(Sum(Zeta(n)^n - 1, n = 2 .. infinity), 120); # Vaclav Kotesovec, Sep 02 2024

RealDigits[N[Sum[Zeta[n]^n - 1, {n, 2, 1000}], 150]][[1]]

cp's OEIS Frontend

A244285 Decimal expansion of A1*B1, the average number of non-isomorphic semisimple rings of any order, where A1 is Product_{m>1} zeta(m) and B1 is Product_{r*m^2 > 1} zeta(r*m^2).

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A082868 Continued fraction approximation to Product_{k>=2} zeta(k).

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A169862 Decimal expansion of root of x^(x-1) = (x-1)^x.

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A369634 Decimal expansion of the infinite product of the Zeta Functions with arguments that are multiples of 3.

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A249141 Decimal expansion of 'sigma', a constant associated with the expected number of random elements to generate a finite abelian group.

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A328705 Dirichlet g.f.: Product_{k>=1} zeta(k*s)^2.

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A335494 Decimal expansion of s, where s is the root of Product_{k>=1} zeta(k*s) = 2.

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A338851 Constant T such that Sum_{n>=1} zeta(T*n)=1.

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A244285 Decimal expansion of A1B1, the average number of non-isomorphic semisimple rings of any order, where A1 is Product_{m>1} zeta(m) and B1 is Product_{rm^2 > 1} zeta(r*m^2).