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%I M0064 N0020 #191 Feb 26 2024 10:42:07
%S 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,5,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,7,1,1,
%T 1,4,1,1,1,3,1,1,1,2,2,1,1,5,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,11,1,1,1,2,
%U 1,1,1,6,1,1,2,2,1,1,1,5,5,1,1,2,1,1,1,3,1,2,1,2,1,1,1,7,1,2,2,4,1,1,1,3,1,1,1
%N Number of Abelian groups of order n; number of factorizations of n into prime powers.
%C Equivalently, number of Abelian groups with n conjugacy classes. - _Michael Somos_, Aug 10 2010
%C a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
%C Also number of rings with n elements that are the direct product of fields; these are the commutative rings with n elements having no nilpotents; likewise the commutative rings where for every element x there is a k > 0 such that x^(k+1) = x. - _Franklin T. Adams-Watters_, Oct 20 2006
%C Range is A033637.
%C a(n) = 1 if and only if n is from A005117 (squarefree numbers). See the Ahmed Fares comment there, and the formula for n>=2 below. - _Wolfdieter Lang_, Sep 09 2012
%C Also, from a theorem of Molnár (see [Molnár]), the number of (non-isomorphic) abelian groups of order 2*n + 1 is equal to the number of non-congruent lattice Z-tilings of R^n by crosses, where a "cross" is a unit cube in R^n for which at each facet is attached another unit cube (Z, R are the integers and reals, respectively). (Cf. [Horak].) - _L. Edson Jeffery_, Nov 29 2012
%C Zeta(k*s) is the Dirichlet generating function of the characteristic function of numbers which are k-th powers (k=1 in A000012, k=2 in A010052, k=3 in A010057, see arXiv:1106.4038 Section 3.1). The infinite product over k (here) is the number of representations n=product_i (b_i)^(e_i) where all exponents e_i are distinct and >=1. Examples: a(n=4)=2: 4^1 = 2^2. a(n=8)=3: 8^1 = 2^1*2^2 = 2^3. a(n=9)=2: 9^1 = 3^2. a(n=12)=2: 12^1 = 3*2^2. a(n=16)=5: 16^1 = 2*2^3 = 4^2 = 2^2*4^1 = 2^4. If the e_i are the set {1,2} we get A046951, the number of representations as a product of a number and a square. - _R. J. Mathar_, Nov 05 2016
%C See A060689 for the number of non-abelian groups of order n. - _M. F. Hasler_, Oct 24 2017
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 274-278.
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.12, p. 468.
%D J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A. Speiser, Die Theorie der Gruppen von endlicher Ordnung, 4. Auflage, Birkhäuser, 1956.
%H T. D. Noe, <a href="/A000688/b000688.txt">Table of n, a(n) for n = 1..10000</a>
%H Tak-Shing T. Chan, and Y.-H. Yang, <a href="https://doi.org/10.1109/TSP.2016.2612171">Polar n-Complex and n-Bicomplex Singular Value Decomposition and Principal Component Pursuit</a>, IEEE Transactions on Signal Processing ( Volume: 64, Issue: 24, Dec.15, 15 2016 ); DOI: 10.1109/TSP.2016.2612171.
%H I. G. Connell, <a href="http://cms.math.ca/10.4153/CMB-1964-002-1">A number theory problem concerning finite groups and rings</a>, Canad. Math. Bull, 7 (1964), 23-34.
%H I. G. Connell, <a href="/A000688/a000688.pdf">Letter to N. J. A. Sloane, no date</a>
%H P. Erdős and G. Szekeres, <a href="http://acta.bibl.u-szeged.hu/13441/1/math_007_095-102.pdf">Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem</a>, Acta Sci. Math. (Szeged), 7 (1935), 95-102.
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/abel/abel.html">Abelian Group Enumeration Constants</a> [Broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010603070928/http://www.mathsoft.com/asolve/constant/abel/abel.html">Abelian Group Enumeration Constants</a> [broken link?] [From the Wayback machine]
%H P. Horak, <a href="http://dx.doi.org/10.2478/v10127-010-0004-y">Error-correcting codes and Minkowski's conjecture</a>, Tatra Mt. Math. Publ., 45 (2010), p. 40.
%H B. Horvat, G. Jaklic and T. Pisanski, <a href="http://arXiv.org/abs/math.CO/0503183">On the number of Hamiltonian groups</a>, arXiv:math/0503183 [math.CO], 2005.
%H D. G. Kendall, R. A. Rankin, <a href="http://dx.doi.org/10.1093/qmath/os-18.1.197">On the number of Abelian groups of a given order</a>, Q. J. Math. 18 (1947) 197-208.
%H Nobushige Kurokawa and Masato Wakayama, <a href="https://doi.org/10.3792/pjaa.78.126">Zeta extensions</a>. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 7, 126--130. <a href="http://www.ams.org/mathscinet-getitem?mr=1930216">MR1930216 (2003h:11112)</a>.
%H E. Molnár, <a href="http://www.bdim.eu/item?id=RLINA_1971_8_51_3-4_177_0">Sui mosaici dello spazio di dimensione n</a>, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 51 (1971), 177-185.
%H H.-E. Richert, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002382776&IDDOC=157448">Über die Anzahl Abelscher Gruppen gegebener Ordnung I</a>, Math. Zeitschr. 56 (1952) 21-32.
%H Marko Riedel, <a href="http://math.stackexchange.com/questions/955649/">Counting Abelian Groups</a>, Mathematics Stack Exchange, October 2014.
%H Laszlo Toth, <a href="http://arxiv.org/abs/1203.6473">A note on the number of abelian groups of a given order</a>, arXiv:1203.6473 [math.NT], (2012).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbelianGroup.html">Abelian Group</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FiniteGroup.html">Finite Group</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KroneckerDecompositionTheorem.html">Kronecker Decomposition Theorem</a>
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F Multiplicative with a(p^k) = number of partitions of k = A000041(k); a(mn) = a(m)a(n) if (m, n) = 1.
%F a(2n) = A101872(n).
%F a(n) = Product_{j = 1..N(n)} A000041(e(j)), n >= 2, if
%F n = Product_{j = 1..N(n)} prime(j)^e(j), N(n) = A001221(n). See the Richert reference, quoting A. Speiser's book on finite groups (in German, p. 51 in words). - _Wolfdieter Lang_, Jul 23 2011
%F In terms of the cycle index of the symmetric group: Product_{q=1..m} [z^{v_q}] Z(S_v) 1/(1-z) where v is the maximum exponent of any prime in the prime factorization of n, v_q are the exponents of the prime factors, and Z(S_v) is the cycle index of the symmetric group on v elements. - _Marko Riedel_, Oct 03 2014
%F Dirichlet g.f.: Sum_{n >= 1} a(n)/n^s = Product_{k >= 1} zeta(ks) [Kendall]. - _Álvar Ibeas_, Nov 05 2014
%F a(n)=2 for all n in A054753 and for all n in A085987. a(n)=3 for all n in A030078 and for all n in A065036. a(n)=4 for all n in A085986. a(n)=5 for all n in A030514 and for all n in A178739. a(n)=6 for all n in A143610. - _R. J. Mathar_, Nov 05 2016
%F A050360(n) = a(A025487(n)). a(n) = A050360(A101296(n)). - _R. J. Mathar_, May 26 2017
%F a(n) = A000001(n) - A060689(n). - _M. F. Hasler_, Oct 24 2017
%F From _Amiram Eldar_, Nov 01 2020: (Start)
%F a(n) = a(A057521(n)).
%F Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = A021002. (End)
%F a(n) = A005361(n) except when n is a term of A046101, since A000041(x) = x for x <= 3. - _Miles Englezou_, Feb 17 2024
%e a(1) = 1 since the trivial group {e} is the only group of order 1, and it is Abelian; alternatively, since the only factorization of 1 into prime powers is the empty product.
%e a(p) = 1 for any prime p, since the only factorization into prime powers is p = p^1, and (in view of Lagrange's theorem) there is only one group of prime order p; it is isomorphic to (Z/pZ,+) and thus Abelian.
%e From _Wolfdieter Lang_, Jul 22 2011: (Start)
%e a(8) = 3 because 8 = 2^3, hence a(8) = pa(3) = A000041(3) = 3 from the partitions (3), (2, 1) and (1, 1, 1), leading to the 3 factorizations of 8: 8, 4*2 and 2*2*2.
%e a(36) = 4 because 36 = 2^2*3^2, hence a(36) = pa(2)*pa(2) = 4 from the partitions (2) and (1, 1), leading to the 4 factorizations of 36: 2^2*3^2, 2^2*3^1*3^1, 2^1*2^1*3^2 and 2^1*2^1*3^1*3^1.
%e (End)
%p with(combinat): readlib(ifactors): for n from 1 to 120 do ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*numbpart(ifactors(n)[2][i][2]) od: printf(`%d,`,ans): od: # _James A. Sellers_, Dec 07 2000
%t f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; Array[f, 107] (* _Robert G. Wilson v_, Sep 22 2006 *)
%t Table[FiniteAbelianGroupCount[n], {n, 200}] (* Requires version 7.0 or later. - _Vladimir Joseph Stephan Orlovsky_, Jul 01 2011 *)
%o (PARI) A000688(n) = {local(f);f=factor(n);prod(i=1,matsize(f)[1],numbpart(f[i,2]))} \\ _Michael B. Porter_, Feb 08 2010
%o (PARI) a(n)=my(f=factor(n)[,2]); prod(i=1,#f,numbpart(f[i])) \\ _Charles R Greathouse IV_, Apr 16 2015
%o (Sage)
%o def a(n):
%o F=factor(n)
%o return prod([number_of_partitions(F[i][1]) for i in range(len(F))])
%o # _Ralf Stephan_, Jun 21 2014
%o (Haskell)
%o a000688 = product . map a000041 . a124010_row
%o -- _Reinhard Zumkeller_, Aug 28 2014
%o (Python)
%o from sympy import factorint, npartitions
%o from math import prod
%o def A000688(n): return prod(map(npartitions,factorint(n).values())) # _Chai Wah Wu_, Jan 14 2022
%Y Cf. A000001, A021002, A060689, A000041, A000961, A001055, A005361, A034382, A046054, A046055, A046056, A046101, A050360, A055653, A057521, A101872 (bisection), A101876 (quadrisection), A124010, A050361, A051532, A129667 (Dirichlet inverse).
%Y Cf. A080729 (Dgf at s=2), A369634 (Dgf at s=3).
%K nonn,core,easy,nice,mult
%O 1,4
%A _N. J. A. Sloane_
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