A084892 -id:A084892 - cp's OEIS frontend

A063966 Number of Abelian groups of order <= n.

1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 17, 18, 19, 20, 25, 26, 28, 29, 31, 32, 33, 34, 37, 39, 40, 43, 45, 46, 47, 48, 55, 56, 57, 58, 62, 63, 64, 65, 68, 69, 70, 71, 73, 75, 76, 77, 82, 84, 86, 87, 89, 90, 93, 94, 97, 98, 99, 100, 102, 103, 104, 106, 117, 118, 119, 120, 122

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Sep 04 2001

nonn

Vaclav Kotesovec, 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.
Hong-Quan Liu, On the number of abelian groups of a given order (supplement), Acta Arithmetica, Vol. 64, No. 3 (1993), pp. 285-296.
Eric Weisstein's World of Mathematics, Abelian Group.

Partial sums of A000688.
Cf. A063756.
Cf. A021002, A084892, A084893.

with(combinat): readlib(ifactors): total := 0: for n from 1 to 100 do ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*numbpart(ifactors(n)[2][i][2]) od: printf(`%d,`,total+ans): total := total+ans: od:

Accumulate[Table[FiniteAbelianGroupCount[n], {n, 1, 200}]] (* Geoffrey Critzer, Dec 28 2014 *)

a(n) ~ c * n, where c = A021002 = Product_{k>=2} zeta(k). - Vaclav Kotesovec, Oct 26 2019

More accurately, a(n) = A021002 * n + A084892 * n^(1/2) + A084893 * n^(1/3) + O(n^(50/199 + eps)), where eps>0 is arbitrarily small (Liu, 1993). - Amiram Eldar, Sep 23 2023

More terms from James Sellers, Sep 26 2001

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

A084911 Decimal expansion of linear asymptotic constant B in Sum_{k=1..n} 1/A000688(k) = ~B*n + ...

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jun 11 2003

			0.7520107423...

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

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See p. 16.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1; arXiv preprint, arXiv:1608.00795 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Abelian Group.

Cf. A000041, A000688, A021002, A084892, A084893, A272339.

digits = 10; m0 (* initial number of primes *) = 10^6; dm = 2*10^5; PP = PartitionsP; DP[n_] := DP[n] = (1/PP[n - 1] - 1 /PP[n]) // N[#, digits + 5]&; pmax = Prime[1000];
nmax[p_ /; p <= pmax] := nmax[p] = Module[{n}, For[n = 2, n < 1000, n++, If[Abs[1/PP[n - 1] - 1 /PP[n]]/p^n < 10^-100, Return[n]]]]; nmax[p_ /; p > pmax] := nmax[pmax];
s[p_] := Sum[DP[n]/p^n, {n, 2, nmax[p]}] ;
f[m_] := f[m] = Product[1 - s[p], {p, Prime[Range[m]]}]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits + 2][[1]] != RealDigits[f[m - dm], 10, digits + 2][[1]], m = m + dm; Print[m, " ", RealDigits[f[m]]]];
A0 = f[m]; RealDigits[A0, 10, digits][[1]] (* Jean-François Alcover, Apr 29 2016 *)

default(realprecision, 120); default(parisize, 10000000);
prodeulerrat((1-1/p)*(1 + sum(i = 1, 512, 1/(numbpart(i)*p^i)))) \\ Amiram Eldar, Mar 08 2024

Equals Product_{p prime} (1-Sum_{k >= 2} (1/P(k-1)-1/P(k))/p^k), where P(k) is the unrestricted partition function. - Jean-François Alcover, Apr 29 2016, [typo corrected by Vaclav Kotesovec, Mar 05 2024]

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} 1/A000688(k). - Amiram Eldar, Oct 16 2020

Data corrected by Jean-François Alcover, Apr 29 2016
a(10) from Vaclav Kotesovec, Mar 07 2024
More terms from Amiram Eldar, Mar 08 2024

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).

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)).

A245055 Decimal expansion of 'tau' (named sigma_2 by C. Pomerance), a constant associated with the expected number of random elements to generate a finite abelian group.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 22 2014

			1.7426523110335154352489048069412986411544379898381...

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, A249141.

digits = 101; max = 400; c = 1/Product[N[Zeta[k], digits + 100], {k, 2, max}]; p[j_] := Product[N[Zeta[k], digits + 100], {k, 2, j}]; tau = Sum[1 - (1 - 2^-j)*c*p[j], {j, 1, max}]; RealDigits[tau, 10, digits ] // First

default(realprecision,120); suminf(j=1, 1-(1-2^(-j))*prodinf(k=j+1, 1/zeta(k))) \\ Vaclav Kotesovec, Oct 22 2014

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

tau = sum_{j >= 1} (1-(1-2^(-j))*c*prod_{k = 2..j} zeta(k)), where c is A068982.

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A063966 Number of Abelian groups of order <= n.

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

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A084911 Decimal expansion of linear asymptotic constant B in Sum_{k=1..n} 1/A000688(k) = ~B*n + ...

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

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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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Mathematica

PARI

Formula

A245055 Decimal expansion of 'tau' (named sigma_2 by C. Pomerance), a constant associated with the expected number of random elements to generate a finite abelian group.

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Author

Keywords

Examples

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Crossrefs

Programs

Mathematica

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Formula