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A001001 Number of sublattices of index n in generic 3-dimensional lattice.

%I A001001 #106 Apr 10 2025 22:04:25
%S A001001 1,7,13,35,31,91,57,155,130,217,133,455,183,399,403,651,307,910,381,
%T A001001 1085,741,931,553,2015,806,1281,1210,1995,871,2821,993,2667,1729,2149,
%U A001001 1767,4550,1407,2667,2379,4805,1723,5187,1893,4655,4030,3871,2257,8463,2850,5642,3991,6405,2863
%N A001001 Number of sublattices of index n in generic 3-dimensional lattice.
%C A001001 These sublattices are in 1-1 correspondence with matrices
%C A001001 [a b d]
%C A001001 [0 c e]
%C A001001 [0 0 f]
%C A001001 with acf = n, b = 0..c-1, d = 0..f-1, e = 0..f-1. The sublattice is primitive if gcd(a,b,c,d,e,f) = 1.
%C A001001 Total area of all distinct rectangles whose side lengths are divisors of n, and whose length is an integer multiple of the width. - _Wesley Ivan Hurt_, Aug 23 2020
%D A001001 Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(d), pp. 76 and 113.
%H A001001 Amiram Eldar, <a href="/A001001/b001001.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A001001 Michael Baake, <a href="https://arxiv.org/abs/math/0605222">Solution of the coincidence problem in dimensions d <= 4</a>, in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.; arXiv:math/0605222 [math.MG], 2006.
%H A001001 M. Baake and N. Neumarker, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Baake/baake7.html">A Note on the Relation Between Fixed Point and Orbit Count Sequences</a>, JIS 12 (2009) 09.4.4, Section 3.
%H A001001 J. Liouville, <a href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1857_2_2_A4_0.pdf">Théorème concernant les sommes de diviseurs des nombres</a>, Journal de mathématiques pures et appliquées 2e série, tome 2 (1857), p. 56.
%H A001001 V. A. Liskovets and A. Mednykh, <a href="https://doi.org/10.1080/00927870008826924">Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces</a>, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.
%H A001001 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A001001 J. S. Rutherford, <a href="https://dx.doi.org/10.1107/S0108767392000898">The enumeration and symmetry-significant properties of derivative lattices</a>, Act. Cryst. (1992) A48, 500-508
%H A001001 J. S. Rutherford, <a href="https://doi.org/10.1107/S0108767392007657">The enumeration and symmetry-significant properties of derivative lattices II</a>, Acta Cryst. A49 (1993), 293-300. [_N. J. A. Sloane_, Mar 14 2009]
%H A001001 Tad White, <a href="http://arxiv.org/abs/1304.2830">Counting Free Abelian Actions</a>, arXiv:1304.2830 [math.CO], 2013.
%H A001001 <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>.
%F A001001 If n = Product p^m, a(n) = Product (p^(m + 1) - 1)(p^(m + 2) - 1)/(p - 1)(p^2 - 1). Or, a(n) = Sum_{d|n} sigma(n/d)*d^2, Dirichlet convolution of A000290 and A000203.
%F A001001 a(n) = Sum_{d|n} d*sigma(d). - _Vladeta Jovovic_, Apr 06 2001
%F A001001 Multiplicative with a(p^e) = ((p^(e+1)-1)(p^(e+2)-1))/((p-1)(p^2-1)). - _David W. Wilson_, Sep 01 2001
%F A001001 Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2).
%F A001001 L.g.f.: -log(Product_{k>=1} (1 - x^k)^sigma(k)) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, May 23 2018
%F A001001 a(n) = Sum_{d1|n, d2|n, d1|d2} d1*d2. - _Wesley Ivan Hurt_, Aug 23 2020
%F A001001 Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2*zeta(3)/18 = 0.659101... . - _Amiram Eldar_, Oct 19 2022
%F A001001 G.f.: Sum_{k>=1} Sum {l>=1} k*l^2*x^(k*l - 1)/(1 - x^(k*l)). - _Miles Wilson_, Apr 04 2025
%p A001001 nmax := 100:
%p A001001 L12 := [seq(1,i=1..nmax) ];
%p A001001 L27 := [seq(i,i=1..nmax) ];
%p A001001 L290 := [seq(i^2,i=1..nmax) ];
%p A001001 DIRICHLET(L12,L27) ;
%p A001001 DIRICHLET(%,L290) ; # _R. J. Mathar_, Sep 25 2017
%t A001001 a[n_] := Sum[ d*DivisorSigma[1, d], {d, Divisors[n]}]; Table[ a[n], {n, 1, 42}] (* _Jean-François Alcover_, Jan 20 2012, after _Vladeta Jovovic_ *)
%t A001001 f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 2}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Aug 29 2019 *)
%o A001001 (PARI)
%o A001001 N=17; default(seriesprecision,N); x=z+O(z^(N+1))
%o A001001 c=sum(j=1,N,j*x^j);
%o A001001 t=1/prod(j=1,N, eta(x^(j))^j)
%o A001001 t=log(t)
%o A001001 t=serconvol(t,c)
%o A001001 Vec(t)
%o A001001 /* _Joerg Arndt_, May 03 2008 */
%o A001001 (PARI) a(n)=sumdiv(n,d, d * sumdiv(d,t, t ) );  /* _Joerg Arndt_, Oct 07 2012 */
%o A001001 (PARI) a(n)=sumdivmult(n,d, sigma(d)*d) \\ _Charles R Greathouse IV_, Sep 09 2014
%o A001001 (Haskell)
%o A001001 a001001 n = sum [sum [k * (if k `mod` l == 0 then l else 0) | k <- [1..n], n `mod` k == 0 ] | l <- [1..n]]
%o A001001 a = [ a001001 n | n <- [1..53]]
%o A001001 putStrLn $ concat $ map (++ ", ") (map show a) -- _Miles Wilson_, Apr 04 2025
%Y A001001 Column 3 of A160870.
%Y A001001 Cf. A060983, A064987 (Mobius transform).
%Y A001001 Cf. A061256, A226313, A301777.
%Y A001001 Primes in this sequence are in A053183.
%Y A001001 Cf. A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.
%K A001001 nonn,easy,nice,mult
%O A001001 1,2
%A A001001 _N. J. A. Sloane_