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A321015 Number of isohedral Voronoi parallelotopes in R^n.

%I A321015 #19 Feb 02 2025 02:28:31
%S A321015 1,2,2,4,3,6,4,7,4,6,3,10,3,7,6,9,3,10,3,10,7,6,3,15,5,6,6,11,3,14,3,
%T A321015 11,6,6,8,16,3,6,6,15,3,15,3,10,10,6,3,19,6,10,6,10,3,14,7,16,6,6,3,
%U A321015 22,3,6,11,13,7,14,3,10,6,15,3,23,3,6,10,10,8,14,3,19,8,6,3,23,7
%N A321015 Number of isohedral Voronoi parallelotopes in R^n.
%H A321015 Amiram Eldar, <a href="/A321015/b321015.txt">Table of n, a(n) for n = 1..10000</a>
%H A321015 Marjorie Senechal, <a href="https://doi.org/10.1007/978-3-662-02838-4_10">Introduction to lattice geometry</a>. In M. Waldschmidt et al., eds., From Number Theory to Physics, pp. 476-495. Springer, Berlin, Heidelberg, 1992. See Cor. 3.7.
%F A321015 a(n) = d(n) + A321013(n) + A321014(n), where d(n) = A000005(n) is the number of divisors of n.
%F A321015 Sum_{k=1..n} a(k) ~ n * (2*log(n) + 4*gamma - 571/168), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Feb 02 2025
%e A321015 Of the five different Voronoi cells of 3-dimensional lattices, only two are isohedral, so a(3) = 2: the cube and the rhombic dodecahedron, the Voronoi cells of the primitive cubic and the face-centered cubic lattices.
%p A321015 d2:=proc(n) local c; if n <= 3 then return(0); fi;
%p A321015 c:=NumberTheory[tau](n)-1;
%p A321015 if (n mod 2)=0 then c:=c-1; fi;
%p A321015 if (n mod 3)=0 then c:=c-1; fi; c; end; # A321014
%p A321015 d3:=proc(n) local c;  c:=0;
%p A321015 if (n mod 6)=0 then c:=c+1; fi;
%p A321015 if (n mod 7)=0 then c:=c+1; fi;
%p A321015 if (n mod 8)=0 then c:=c+1; fi; c; end; # A321013
%p A321015 [seq(NumberTheory[tau](n)+d2(n)+d3(n),n=1..120)];
%o A321015 (PARI) a(n) = 2*numdiv(n) + sum(k = 6, 8, !(n % k)) + n%2 + (n%3>0) - 3; \\ _Amiram Eldar_, Feb 02 2025
%Y A321015 Cf. A000005, A001620, A321013, A321014, A071880, A071881, A071882.
%K A321015 nonn
%O A321015 1,2
%A A321015 _N. J. A. Sloane_, Nov 04 2018