cp's OEIS Frontend

A152455 a(n) = minimal integer m such that there exists an m X m integer matrix of order n.

%I A152455 #15 Apr 04 2023 10:46:39
%S A152455 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 A152455 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 A152455 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 A152455 a(n) = minimal integer m such that there exists an m X m integer matrix of order n.
%C A152455 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 A152455 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 A152455 Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, 1985, p. 51.
%H A152455 Michael De Vlieger, <a href="/A152455/b152455.txt">Table of n, a(n) for n = 1..10000</a>
%H A152455 Howard Hiller, <a href="https://doi.org/10.1107/S0108767385001180">The Crystallographic Restriction in Higher Dimensions</a>, Acta Cryst. (1985), A41, 541-544.
%H A152455 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 A152455 a(1)=0, a(2)=1. If n mod 4 eq 2 then a(n)=a(n/2).
%F A152455 Otherwise a(n) = sum (pi-1)*pi^(ei-1) where n = p1^e1*p2^e2*...pk^ek is prime factorization of n.
%t A152455 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 A152455 (Magma) a := function(n)
%o A152455 if n le 2 then return n-1; end if;
%o A152455 if n mod 4 eq 2 then n := n div 2; end if;
%o A152455 f := Factorization(n);
%o A152455 return &+[(t[1]-1)*t[1]^(t[2]-1):t in f];
%o A152455 end function;
%Y A152455 See A080737 for another version. - _N. J. A. Sloane_, Dec 05 2008
%K A152455 easy,nonn
%O A152455 1,3
%A A152455 W. R. Unger (billu(AT)maths.usyd.edu.au), Dec 04 2008