A063966 -id:A063966 - cp's OEIS frontend

A021002 Decimal expansion of zeta(2)zeta(3)zeta(4)*...

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

Offset: 1

Andre Neumann Kauffman (ank(AT)nlink.com.br)

A very good approximation is 2e-Pi = ~2.29497100332829723225793155942... - Marco Matosic, Nov 16 2005

This constant is equal to the asymptotic mean of number of Abelian groups of order n (A000688). - Amiram Eldar, Oct 16 2020

			2.2948565916733137941835158313443112887131637994416686732758140300...

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963, p. 198-9.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.

Robert Price, Table of n, a(n) for n = 1..1000
Steven R. Finch, Abelian Group Enumeration Constants. [From the Wayback machine]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 86.
Felix Fontein and Pawel Wocjan, Quantum Algorithm for Computing the Period Lattice of an Infrastructure, arXiv preprint arXiv:1111.1348 [quant-ph], 2011.
Felix Fontein and Pawel Wocjan, On the probability of generating a lattice, Journal of Symbolic Computation, Vol. 64 (2014), pp. 3-15, arXiv preprint, arXiv:1211.6246 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Jan 03 2013
Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers, Vol. 9 (2009), Article #A08, pp. 83-106; alternative link; arXiv:0604505 [math.NT], 2006.
B. R. Srinivasan, On the Number of Abelian Groups of a Given Order, Acta Arithmetica, Vol. 23, No. 2 (1973), pp. 195-205, alternative link.
Eric Weisstein's World of Mathematics, Abelian Group.
Index entries for constants related to zeta

Cf. A068982 (reciprocal), A082868 (continued fraction).

Cf. A002117, A000688, A063966, A080729, A080730.

evalf(product(Zeta(n), n=2..infinity), 200);

p = Product[ N[ Zeta[n], 256], {n, 2, 1000}]; RealDigits[p, 10, 111][[1]] (* Robert G. Wilson v, Nov 22 2005 *)

prodinf(n=2,zeta(n)) \\ Charles R Greathouse IV, May 27 2015

Product of A080729 and A080730. - R. J. Mathar, Feb 16 2011

More terms from Simon Plouffe, Jan 07 2002
Further terms from Robert G. Wilson v, Nov 22 2005
Mathematica program fixed by Vaclav Kotesovec, Sep 20 2014

A084892 Decimal expansion of Product_{j>=1, j!=2} zeta(j/2) (negated).

Original entry on oeis.org

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

Offset: 2

Eric W. Weisstein, Jun 10 2003

This constant, A_2, appears in the asymptotic formula A063966(n) = Sum_{k=1..n} A000688(k) = A_1 * n + A_2 * n^(1/2) + A_3 * n^(1/3) + O(n^(50/199 + e)), where e>0 is arbitrarily small, A_1 = A021002, and A_3 = A084893. - Amiram Eldar, Oct 16 2020

			-14.64756630163831131699976...

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

B. R. Srinivasan, On the Number of Abelian Groups of a Given Order, Acta Arithmetica, Vol. 23, No. 2 (1973), pp. 195-205, alternative link.
Eric Weisstein's World of Mathematics, Abelian Group.

Cf. A000688, A021002, A063966, A084893.

m0 = 100; dm = 100; digits = 102; Clear[p]; p[m_] := p[m] = Zeta[1/2]*Product[Zeta[j/2], {j, 3, m}]; p[m0]; p[m = m0 + dm]; While[RealDigits[p[m], 10, digits + 10] != RealDigits[p[m - dm], 10, digits + 10], Print["m = ", m]; m = m + dm]; RealDigits[p[m], 10, digits] // First (* Jean-François Alcover, Jun 23 2014 *)

prodinf(k=1, if (k!=2, zeta(k/2), 1)) \\ Michel Marcus, Oct 16 2020

A084893 Decimal expansion of Product_{j>=1, j!=3} zeta(j/3).

Original entry on oeis.org

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

Offset: 3

Eric W. Weisstein, Jun 10 2003

This constant, A_3, appears in the asymptotic formula A063966(n) = Sum_{k=1..n} A000688(k) = A_1 * n + A_2 * n^(1/2) + A_3 * n^(1/3) + O(n^(50/199 + e)), where e>0 is arbitrarily small, A_1 = A021002, and A_2 = A084892. - Amiram Eldar, Oct 16 2020

			118.6924619727642846261669938137118...

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

B. R. Srinivasan, On the Number of Abelian Groups of a Given Order, Acta Arithmetica, Vol. 23, No. 2 (1973), pp. 195-205, alternative link.
Eric Weisstein's World of Mathematics, Abelian Group.

Cf. A000688, A021002, A063966, A084892.

m0 = 100; dm = 100; digits = 102; Clear[p]; p[m_] := p[m] = Zeta[1/3]*Zeta[2/3]*Product[Zeta[j/3], {j, 4, m}]; p[m0]; p[m = m0 + dm]; While[RealDigits[p[m], 10, digits + 10] != RealDigits[p[m - dm], 10, digits + 10], Print["m = ", m]; m = m + dm]; RealDigits[p[m], 10, digits] // First (* Jean-François Alcover, Jun 23 2014 *)

prodinf(k=1, if (k!=3, zeta(k/3), 1)) \\ Michel Marcus, Oct 16 2020

A370897 Partial alternating sums of the number of abelian groups sequence (A000688).

Original entry on oeis.org

1, 0, 1, -1, 0, -1, 0, -3, -1, -2, -1, -3, -2, -3, -2, -7, -6, -8, -7, -9, -8, -9, -8, -11, -9, -10, -7, -9, -8, -9, -8, -15, -14, -15, -14, -18, -17, -18, -17, -20, -19, -20, -19, -21, -19, -20, -19, -24, -22, -24, -23, -25, -24, -27, -26, -29, -28, -29, -28

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000688, A063966.
Cf. A021002, A048651, A084892, A084893.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

f[n_] := Times @@ (PartitionsP[Last[#]] & /@ FactorInteger[n]); f[1] = 1; Accumulate[Array[(-1)^(#+1) * f[#] &, 100]]

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * f(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A000688(k).

a(n) = k_1 * A021002 * n + k_2 * A084892 * n^(1/2) + k_3 * A084893 * n^(1/3) + O(n^(1/4 + eps)), where eps > 0 is arbitrarily small, k_j = -1 + 2 * Product_{i>=1} (1 - 1/2^(i/j)), k_1 = 2*A048651 - 1 = -0.422423809826..., k_2 = -0.924973966404..., and k_3 = -0.991478298912... (Tóth, 2017).

A104452 Number of groups of order <= n all of whose subgroups are normal.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 12, 14, 15, 16, 18, 19, 20, 21, 27, 28, 30, 31, 33, 34, 35, 36, 40, 42, 43, 46, 48, 49, 50, 51, 59, 60, 61, 62, 66, 67, 68, 69, 73, 74, 75, 76, 78, 80, 81, 82, 88, 90, 92, 93, 95, 96, 99, 100, 104, 105, 106, 107, 109, 110, 111, 113, 125, 126, 127

Offset: 1

Boris Horvat (Boris.Horvat(AT)fmf.uni-lj.si), Gasper Jaklic (Gasper.Jaklic(AT)fmf.uni-lj.si), Tomaz Pisanski, Apr 19 2005

Robert D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.
John C. Lennox and Stewart. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Boris Horvat, Gašper Jaklič, and Tomaž Pisanski, On the number of hamiltonian groups, Mathematical Communications, Vol. 10, No. 1 (2005), pp. 89-94; arXiv preprint, arXiv:math/0503183 [math.CO], 2005.
Tomaž Pisanski and Thomas W. Tucker, The genus of low rank hamiltonian groups, Discrete Math. 78 (1989), 157-167.
Eric Weisstein's World of Mathematics, Abelian Group.
Eric Weisstein's World of Mathematics, Hamiltonian Group.

Partial sums of A104404.

Cf. A000688, A021002, A048651, A063966, A104488, A104407, A104453.

orders[n_]:=Map[Last, FactorInteger[n]]; a[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/;Mod[n, 8]==0:=a[e[n]]; h[n_]:=0; numberOfAbelianGroupsOfOrderLEQThanN[n_]:=Map[Apply[Plus, # ]&, Table[Take[Map[a, Table[i, {i, 1, n}]], i], {i, 1, n}]]; numberOfHamiltonianGroupsOfOrderLEQThanN[n_]:=Map[Apply[Plus, # ]&, Table[Take[Map[h, Table[i, {i, 1, n}]], i], {i, 1, n}]]; numberOfAllGroupsOfOrderLEQThanN[n_]:=numberOfAbelianGroupsOfOrderLEQThanN[n] +numberOfHamiltonianGroupsOfOrderLEQThanN[n];

a(n) ~ c * n, where c = A021002 * (1 + A048651/4) = 2.46053840757488111675... . - Amiram Eldar, Oct 03 2023

A104407 Number of Hamiltonian groups of order <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12

Offset: 1

Boris Horvat (Boris.Horvat(AT)fmf.uni-lj.si), Gasper Jaklic (Gasper.Jaklic(AT)fmf.uni-lj.si), Tomaz Pisanski, Apr 19 2005

Robert D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.
John C. Lennox and Stewart. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Boris Horvat, Gašper Jaklič, and Tomaž Pisanski, On the number of hamiltonian groups, Mathematical Communications, Vol. 10, No. 1 (2005), pp. 89-94; arXiv preprint, arXiv:math/0503183 [math.CO], 2005.
Tomaž Pisanski and Thomas W. Tucker, The genus of low rank hamiltonian groups, Discrete Math. 78 (1989), 157-167.
Eric Weisstein's World of Mathematics, Hamiltonian Group.

Partial sums of A104488.

Cf. A000688, A021002, A048651, A063966, A104404, A104452, A104453.

orders[n_]:=Map[Last, FactorInteger[n]]; a[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/;Mod[n, 8]==0:=a[e[n]]; h[n_]:=0; numberOfHamiltonianGroupsOfOrderLEQThanN[n_]:=Map[Apply[Plus, # ]&, Table[Take[Map[h, Table[i, {i, 1, n}]], i], {i, 1, n}]];

a(n) ~ c * n, where c = A021002 * A048651 / 4 = 0.16568181590156732257... . - Amiram Eldar, Oct 03 2023

A104453 Smallest order for which there are n nonisomorphic finite Hamiltonian groups, or 0 if no such order exists.

Original entry on oeis.org

8, 72, 216, 1800, 648, 5400, 1944, 88200, 27000, 16200, 10, 5832, 264600, 0, 48600, 17496, 10672200, 0, 1323000, 0, 793800, 20, 243000, 52488, 0, 32016600, 405000, 0, 9261000, 2381400, 0, 157464

Offset: 1

Boris Horvat (Boris.Horvat(AT)fmf.uni-lj.si), Gasper Jaklic (Gasper.Jaklic(AT)fmf.uni-lj.si), Tomaz Pisanski, Apr 19 2005

R. D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.
J. C. Lennox and S. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.

B. Horvat, G. Jaklic and T. Pisanski, On the number of Hamiltonian groups, arXiv:math/0503183 [math.CO], 2005.
T. Pisanski and T.W. Tucker, The genus of low rank hamiltonian groups, Discrete Math. 78 (1989), 157-167.
Eric Weisstein's World of Mathematics, Abelian Group
Eric Weisstein's World of Mathematics, Hamiltonian Group

Cf. A000688, A063966, A104488, A104407, A104404, A104452.

S_h(n) denotes the smallest number k for which exactly n nonisomorphic hamiltonian groups of order k exist. Here 0 indicates the case when n is not a product of partition numbers and S_h(n) does not exist.

A379359 Numerators of the partial sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 2, 3, 7, 9, 11, 13, 41, 22, 25, 28, 59, 65, 71, 77, 391, 421, 218, 233, 481, 511, 541, 571, 581, 298, 313, 106, 217, 227, 237, 247, 1739, 1809, 1879, 1949, 3933, 4073, 4213, 4353, 13199, 13619, 14039, 14459, 14669, 14879, 15299, 15719, 15803, 16013, 16223, 16643

Offset: 1

Amiram Eldar, Dec 21 2024

			Fractions begin with 1, 2, 3, 7/2, 9/2, 11/2, 13/2, 41/6, 22/3, 25/3, 28/3, 59/6, ...

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 13-16, Theorem 1.3.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See section 5.1, Abelian group enumeration constants, p. 274.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.8, pp. 27-28.

Cf. A000688, A063966, A084911, A370897, A379360 (denominators), A379361.

Numerator[Accumulate[Table[1/FiniteAbelianGroupCount[n], {n, 1, 100}]]]

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / f(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A000688(k)).

a(n)/A379360(n) = D * n + O(sqrt(n/log(n))), where D = A084911.

A379360 Denominators of the partial sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 3, 3, 3, 6, 6, 6, 6, 30, 30, 15, 15, 30, 30, 30, 30, 30, 15, 15, 5, 10, 10, 10, 10, 70, 70, 70, 70, 140, 140, 140, 140, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 140, 140, 140, 140, 140, 140, 140, 140

Offset: 1

Amiram Eldar, Dec 21 2024

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 13-16, Theorem 1.3.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See section 5.1, Abelian group enumeration constants, p. 274.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.8, pp. 27-28.

Cf. A000688, A063966, A370897, A379359 (numerators), A379362.

Denominator[Accumulate[Table[1/FiniteAbelianGroupCount[n], {n, 1, 100}]]]

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / f(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A000688(k)).

A379361 Numerators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 0, 1, 1, 3, 1, 3, 7, 5, 2, 5, 7, 13, 7, 13, 59, 89, 37, 52, 89, 119, 89, 119, 109, 62, 47, 52, 89, 119, 89, 119, 803, 1013, 803, 1013, 1921, 2341, 1921, 2341, 2201, 2621, 2201, 2621, 2411, 2621, 2201, 2621, 2537, 2747, 2537, 2957, 2747, 3167, 1009, 1149, 3307

Offset: 1

Amiram Eldar, Dec 21 2024

			Fractions begin with 1, 0, 1, 1/2, 3/2, 1/2, 3/2, 7/6, 5/3, 2/3, 5/3, 7/6, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.8, pp. 27-28.

Cf. A000041, A000688, A063966, A084911, A370897, A379359, A379362 (denominators).

Numerator[Accumulate[Table[(-1)^(n+1)/FiniteAbelianGroupCount[n], {n, 1, 100}]]]

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / f(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A000688(k)).

a(n)/A379362(n) ~ D * c * n, where D = A084911, c = 2/(1 + Sum_{k>=1} 1/(P(k)*2^k)) - 1 = 0.18634377034863729099..., and P(k) = A000041(k).

cp's OEIS Frontend

A021002 Decimal expansion of zeta(2)*zeta(3)*zeta(4)*...

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A084892 Decimal expansion of Product_{j>=1, j!=2} zeta(j/2) (negated).

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A084893 Decimal expansion of Product_{j>=1, j!=3} zeta(j/3).

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A370897 Partial alternating sums of the number of abelian groups sequence (A000688).

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A104452 Number of groups of order <= n all of whose subgroups are normal.

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A104407 Number of Hamiltonian groups of order <= n.

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A104453 Smallest order for which there are n nonisomorphic finite Hamiltonian groups, or 0 if no such order exists.

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A379359 Numerators of the partial sums of the reciprocals of the number of abelian groups function (A000688).

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A021002 Decimal expansion of zeta(2)zeta(3)zeta(4)*...