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 A001001 Number of sublattices of index n in generic 3-dimensional lattice. 45
 1, 7, 13, 35, 31, 91, 57, 155, 130, 217, 133, 455, 183, 399, 403, 651, 307, 910, 381, 1085, 741, 931, 553, 2015, 806, 1281, 1210, 1995, 871, 2821, 993, 2667, 1729, 2149, 1767, 4550, 1407, 2667, 2379, 4805, 1723, 5187, 1893, 4655, 4030, 3871, 2257, 8463, 2850, 5642, 3991, 6405, 2863 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS These sublattices are in 1-1 correspondence with matrices [a b d] [0 c e] [0 0 f] 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. Equals row sums of triangle A127108. - Gary W. Adamson, Jul 27 2008 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 REFERENCES M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44. R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(d), pp. 76 and 113. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe) M. Baake, N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3. J. Liouville, Théorème concernant les sommes de diviseurs des nombres, Journal de mathématiques pures et appliquées 2e série, tome 2 (1857), p. 56-. V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738. Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. (1992) A48, 500-508 J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices II, Acta Cryst. A49 (1993), 293-300. [N. J. A. Sloane, Mar 14 2009] Tad White, Counting Free Abelian Actions, arXiv:1304.2830 [math.CO], 2013. FORMULA 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. a(n) = Sum_{d|n} d*sigma(d). - Vladeta Jovovic, Apr 06 2001 Multiplicative with a(p^e) = ((p^(e+1)-1)(p^(e+2)-1))/((p-1)(p^2-1)). - David W. Wilson, Sep 01 2001 Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2). 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 a(n) = Sum_{d1|n, d2|n, d1|d2} d1*d2. - Wesley Ivan Hurt, Aug 23 2020 MAPLE nmax := 100: L12 := [seq(1, i=1..nmax) ]; L27 := [seq(i, i=1..nmax) ]; L290 := [seq(i^2, i=1..nmax) ]; DIRICHLET(L12, L27) ; DIRICHLET(%, L290) ; # R. J. Mathar, Sep 25 2017 MATHEMATICA 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 *) 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 *) PROG (PARI) N=17; default(seriesprecision, N); x=z+O(z^(N+1)) c=sum(j=1, N, j*x^j); t=1/prod(j=1, N, eta(x^(j))^j) t=log(t) t=serconvol(t, c) Vec(t) /* Joerg Arndt, May 03 2008 */ (PARI) a(n)=sumdiv(n, d, d * sumdiv(d, t, t ) );  /* Joerg Arndt, Oct 07 2012 */ (PARI) a(n)=sumdivmult(n, d, sigma(d)*d) \\ Charles R Greathouse IV, Sep 09 2014 CROSSREFS Cf. A060983, A064987 (Mobius transform). Cf. A061256, A127108, A226313, A301777. Primes in this sequence are in A053183. Cf. A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999. Sequence in context: A061204 A334783 A060983 * A067692 A117706 A066673 Adjacent sequences:  A000998 A000999 A001000 * A001002 A001003 A001004 KEYWORD nonn,easy,nice,mult AUTHOR STATUS approved

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Last modified September 20 16:50 EDT 2021. Contains 347586 sequences. (Running on oeis4.)