A152455 - OEIS

%I #15 Apr 04 2023 10:46:39

%S 0,1,2,2,4,2,6,4,6,4,10,4,12,6,6,8,16,6,18,6,8,10,22,6,20,12,18,8,28,

%T 6,30,16,12,16,10,8,36,18,14,8,40,8,42,12,10,22,46,10,42,20,18,14,52,

%U 18,14,10,20,28,58,8,60,30,12,32,16,12,66,18,24,10,70,10,72,36,22,20,16,14

%N a(n) = minimal integer m such that there exists an m X m integer matrix of order n.

%C Also lowest dimension in which rotational symmetry of order n is possible for an infinite regular set of points (previously A028496). - _Sean A. Irvine_, Feb 02 2020

%D J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 935 (note has erroneous value of a(11)).

%D Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, 1985, p. 51.

%H Michael De Vlieger, <a href="/A152455/b152455.txt">Table of n, a(n) for n = 1..10000</a>

%H Howard Hiller, <a href="https://doi.org/10.1107/S0108767385001180">The Crystallographic Restriction in Higher Dimensions</a>, Acta Cryst. (1985), A41, 541-544.

%H Savinien Kreczman, Luca Prigioniero, Eric Rowland, and Manon Stipulanti, <a href="https://orbi.uliege.be/bitstream/2268/300422/1/paper-KPRS-submission.pdf">Magic numbers in periodic sequences</a>, Univ. Liège (Belgium, 2023). See p. 7.

%F a(1)=0, a(2)=1. If n mod 4 eq 2 then a(n)=a(n/2).

%F Otherwise a(n) = sum (pi-1)*pi^(ei-1) where n = p1^e1*p2^e2*...pk^ek is prime factorization of n.

%t Array[Set[a[#], # - 1] &, 2]; a[n_] := If[Mod[n, 4] == 2, a[n/2], Total@ Map[(#1 - 1)*#1^(#2 - 1) & @@ # &, FactorInteger[n]]]; Array[a, 120] (* _Michael De Vlieger_, Apr 04 2023 *)

%o (Magma) a := function(n)

%o if n le 2 then return n-1; end if;

%o if n mod 4 eq 2 then n := n div 2; end if;

%o f := Factorization(n);

%o return &+[(t[1]-1)*t[1]^(t[2]-1):t in f];

%o end function;

%Y See A080737 for another version. - _N. J. A. Sloane_, Dec 05 2008

%K easy,nonn

%O 1,3

%A W. R. Unger (billu(AT)maths.usyd.edu.au), Dec 04 2008