A021002 -id:A021002 - cp's OEIS frontend

A101876 Number of Abelian groups of order 4n.

2, 3, 2, 5, 2, 3, 2, 7, 4, 3, 2, 5, 2, 3, 2, 11, 2, 6, 2, 5, 2, 3, 2, 7, 4, 3, 6, 5, 2, 3, 2, 15, 2, 3, 2, 10, 2, 3, 2, 7, 2, 3, 2, 5, 4, 3, 2, 11, 4, 6, 2, 5, 2, 9, 2, 7, 2, 3, 2, 5, 2, 3, 4, 22, 2, 3, 2, 5, 2, 3, 2, 14, 2, 3, 4, 5, 2, 3, 2, 11, 10, 3, 2, 5, 2, 3, 2, 7, 2, 6, 2, 5, 2, 3, 2, 15, 2, 6, 4, 10, 2

Offset: 1

N. J. A. Sloane, Jan 28 2005

Antti Karttunen, Table of n, a(n) for n = 1..65537

Bisection of A101872, quadrisection of A000688.

Cf. A000041, A021002, A048651.

a[n_] := FiniteAbelianGroupCount[4*n]; Array[a, 100] (* Amiram Eldar, Sep 23 2023*)

A101876(n) = factorback(apply(e -> numbpart(e),factor(4*n)[,2])); \\ Antti Karttunen, Sep 27 2018

a(n) = A000688(4*n). - Antti Karttunen, Sep 27 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (4 - 6 * A048651) * A021002 = 5.20306278505563943501... . - Amiram Eldar, Sep 23 2023

More terms from Joshua Zucker, May 10 2006

A104488 Number of Hamiltonian groups of order n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

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.

T. D. Noe, 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.

Cf. A000265, A000688, A021002, A048651, A104404, 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;
(* Second program: *)
a[n_] := If[Mod[n, 8]==0, FiniteAbelianGroupCount[n/2^IntegerExponent[n, 2]], 0]; Array[a, 102] (* Jean-François Alcover, Sep 14 2019 *)

a(n)={my(e=valuation(n, 2)); if(e<3, 0, my(f=factor(n/2^e)[, 2]); prod(i=1, #f, numbpart(f[i])))} \\ Andrew Howroyd, Aug 08 2018

Let n = 2^e*o, where e = e(n) >= 0 and o = o(n) is an odd number. The number h(n) of Hamiltonian groups of order n is given by h(n) = 0, if e(n) < 3 and h(n) = a(o(n)), otherwise, where a(n) = A000688(n) denotes the number of Abelian groups of order n.
a(8*n) = A000688(A000265(n)), a(n) = 0 for n mod 8 <> 0. - Andrew Howroyd, Aug 08 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A021002 * A048651 / 4 = 0.16568181590156732257... . - Amiram Eldar, Sep 23 2023

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).

A068982 Decimal expansion of the limit of the product of a modified zeta function.

Original entry on oeis.org

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

Offset: 0

Andre Neumann Kauffman (andrekff(AT)hotmail.com), Apr 01 2002

The "modified zeta function" Zetam(n) = sum(mu(k)/k^n) may be helpful when searching for a closed form for Apery's constant.

			0.43575707...

Robert Price, Table of n, a(n) for n = 0..999
Melanie Matchett Wood, Probability theory for random groups arising in number theory, arXiv:2301.09687 [math.NT], 2023. See Theorem 3.6 at p. 21.

Cf. A021002, A002117.

with(numtheory); evalf(Product(Sum('mobius(k)/k^n','k'=1..infinity),n=2..infinity),40); Note: For practical reasons you should change "infinity" to some finite value.
evalf(product(1/Zeta(n), n=2..infinity), 120); # Vaclav Kotesovec, Oct 22 2014

digits = 104; 1/NProduct[ Zeta[n], {n, 2, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> 1000] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013 *)

Equals Product_{k=1..oo} Sum_{n=2..oo} mu(k)/k^n.

Equals 1/A021002. - R. J. Mathar, Jan 31 2009

Corrected and extended by R. J. Mathar, Jan 31 2009

Example corrected by R. J. Mathar, Jul 23 2009

A101871 Number of Abelian groups of order 2n+1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 0

N. J. A. Sloane, Jan 28 2005

Bisection of A000688.

Cf. A021002, A048651.

a[n_] := FiniteAbelianGroupCount[2*n + 1]; Array[a, 100, 0] (* Amiram Eldar, Sep 23 2023 *)

From Amiram Eldar, Sep 23 2023: (Start)
a(n) = A000688(2*n+1) = A000688(4*n+2).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * A048651 * A021002 = 1.32545452721253858057... . (End)

More terms from Joshua Zucker, May 10 2006

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

A361179 a(n) = sigma(n)^4.

Original entry on oeis.org

1, 81, 256, 2401, 1296, 20736, 4096, 50625, 28561, 104976, 20736, 614656, 38416, 331776, 331776, 923521, 104976, 2313441, 160000, 3111696, 1048576, 1679616, 331776, 12960000, 923521, 3111696, 2560000, 9834496, 810000, 26873856, 1048576, 15752961, 5308416

Offset: 1

Vaclav Kotesovec, Mar 03 2023

In general, for k>=1, Sum_{m=1..n} sigma(m)^k ~ c(k) * z(k) * n^(k+1) / (k+1), where z(k) = Product_{j=2..k+1} zeta(j).
z(k) tends to A021002 = 2.29485659167331379418351583... if k tends to infinity.
Table of logarithms of the first twenty constants c(k):
log(c1)  = 0
log(c2)  = 0.4185904294034097177091498674425959208785022862606440306200960821...
log(c3)  = 1.0423888168104400391462790418324165821902123159643681963298587386...
log(c4)  = 1.7991790110714031081639242851527957388041981665455193670488985855...
log(c5)  = 2.6531418047626712704435945717713008165192112256395129469527055461...
log(c6)  = 3.5826667694785981489341382260447390026333883927530294731356708082...
log(c7)  = 4.5733843557245275039380976990636718508529417039225677910093512418...
log(c8)  = 5.6152065176325962438798772352645945078887296036246579568363264836...
log(c9)  = 6.7007695219862872061684609152917692899880931107656334442026270254...
log(c10) = 7.8245175718301572361518558972457980392624870372412384620464547480...
log(c11) = 8.9821318589248960303876549202030018215854310738197659104984082438...
log(c12) = 10.170161510396427442300796140752106239603402200741405656518889304...
log(c13) = 11.385778844373902103940190311048453116470874526205115584130363228...
log(c14) = 12.626614423444098003503814842580453502016287945932183786430620101...
log(c15) = 13.890644760144907314506933347339629337810929043024214330654043796...
log(c16) = 15.176115136560648867246990011975416479066956527530401883224856531...
log(c17) = 16.481485806132270823150284520463000397265757050340939883069076823...
log(c18) = 17.805393674783928883671133007206209125657866860089528876021281793...
log(c19) = 19.146624201995507049618714377273936711664382470319966849198205155...
log(c20) = 20.504090088752226662590920186246482636058069128320785639131816842...
c1 = 1, c2 = 5/(2*zeta(2)) = 15/Pi^2.

Cf. A000203, A072861, A361147.

Cf. A361132, A361148.

Mathematica
```
Table[DivisorSigma[1, n]^4, {n, 1, 50}]
```
PARI
```
a(n) = sigma(n)^4;
```

for(n=1, 100, print1(direuler(p=2, n, (1 + p^2*X)*(1 + 3*p*X + 4*p^2*X + 3*p^3*X + p^4*X^2)/((1 - X)*(1 - p*X)*(1 - p^2*X)*(1 - p^3*X)*(1 - p^4*X)))[n], ", "))

Multiplicative with a(p^e) = ((p^(e+1)-1)/(p-1))^4.
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) * zeta(s-4) * Product_{primes p} (1 + 1/p^(3*s-6) + 3/p^(2*s-3) + 5/p^(2*s-4) + 3/p^(2*s-5) + 3/p^(s-1) + 5/p^(s-2) + 3/p^(s-3)).
Sum_{k=1..n} a(k) ~ c * Pi^6 * zeta(3) * zeta(5) * n^5 / 2700, where c = Product_{primes p} (1 + 3/p^2 + 5/p^3 + 3/p^4 + 3/p^5 + 5/p^6 + 3/p^7 + 1/p^9) = 6.0446828090651437986928739783339791032197283386377841627594461874871547391...
a(n) = A000583(A000203(n)).

A080730 Decimal expansion of the infinite product of zeta functions for odd arguments >= 3.

Original entry on oeis.org

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

Offset: 1

Deepak R. N (deepak_rn(AT)safe-mail.net), Mar 08 2003

			1.2602057107052417107678172260024106280343...

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.

Cf. A021002, A080729.

RealDigits[ Product[ Zeta[ 2n + 1], {n, 500}], 10, 110][[1]] (* Robert G. Wilson v, Nov 21 2014 *)

prodinf(x=1, zeta(2*x+1)) \\ Michel Marcus, Nov 22 2014

Decimal expansion of zeta(3)*zeta(5)*zeta(7)*...*zeta(2k+1)*...

Equals A021002/A080729. - Amiram Eldar, Jan 31 2024

More terms from Benoit Cloitre, Mar 08 2003

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

A192005 Number of non-cyclic abelian groups of finite order. The order is given by A013929.

Original entry on oeis.org

1, 2, 1, 1, 4, 1, 1, 2, 1, 2, 1, 6, 3, 2, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 10, 1, 5, 1, 1, 4, 4, 1, 2, 1, 1, 6, 1, 1, 3, 2, 5, 4, 1, 1, 2, 1, 1, 2, 1, 14, 1, 2, 2, 1, 9, 1, 1, 1, 2, 1, 1, 6, 4, 1, 2, 1, 1, 1, 1, 4, 3, 2, 1, 2, 10, 3, 1, 5, 1, 1, 4, 1, 8, 1, 6, 3, 1, 2, 1, 1, 4, 1, 6, 1, 1, 2, 2, 3, 21, 1, 1, 2, 1, 2, 4, 1, 1, 1, 2

Offset: 1

Wolfdieter Lang, Jul 28 2011

Every abelian group of finite order is the direct product of cyclic groups (there may be only one factor). See, e.g., the A. Speiser reference, Satz 43, p. 49, in combination with Satz 42, p. 47, and also Satz 4, p. 17, with the remark on the direct product on page 28.

See the list of abelian groups of small order in the Wikipedia link.

			n=1: there is one abelian group of order 4=A013929(1), which is not the cyclic group Z_4 (in additive notation), namely the Klein 4-group: Z_2 x Z_2 (also denoted by (Z_2)^2).
n=2: there are 2 non-cyclic abelian groups of order 8=A013929(2), namely Z_2 x Z_4 and (Z_2)^3.
n=3: order 9=A013929(3), (Z_3)^2.
n=4: order 12, Z_3 x (Z_2)^2 (note that Z_6 = Z_3 x Z_2 and Z_12 = Z_4 x Z_3, where = means 'is isomorphic to').
n=5: order 16. The four non-cyclic groups are (Z_2)^4, Z_4 x (Z_2)^2, Z_8 x Z_2 and (Z_4)^2.

Andreas Speiser, Die Theorie der Gruppen von endlicher Ordnung, Vierte Auflage, Birkhäuser, 1956.

Wikipedia, List of small groups.

Cf. A000688, A013929, A013661, A021002.

FiniteAbelianGroupCount /@ Select[Range[300], ! SquareFreeQ[#] &] - 1 (* Amiram Eldar, Oct 01 2023 *)

a(n) = A000688(A013929(n)) - 1, n>=1.
See the formula for A000688 using the product of the number of partitions of the exponents in the prime number factorization.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2) * c - 1)/(zeta(2) - 1) - 1 = 3.3025914257..., where c = A021002. - Amiram Eldar, Oct 01 2023

cp's OEIS Frontend

A101876 Number of Abelian groups of order 4n.

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A104488 Number of Hamiltonian groups of order n.

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

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A068982 Decimal expansion of the limit of the product of a modified zeta function.

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Maple

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A101871 Number of Abelian groups of order 2n+1.

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

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A361179 a(n) = sigma(n)^4.

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PARI

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A080730 Decimal expansion of the infinite product of zeta functions for odd arguments >= 3.

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