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

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

Offset: 1

Author

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

Keywords

• easy
• nonn

Comments

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

References

• 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)).
• Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, 1985, p. 51.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Howard Hiller, The Crystallographic Restriction in Higher Dimensions, Acta Cryst. (1985), A41, 541-544.
• Savinien Kreczman, Luca Prigioniero, Eric Rowland, and Manon Stipulanti, Magic numbers in periodic sequences, Univ. Liège (Belgium, 2023). See p. 7.

Crossrefs

See A080737 for another version. - N. J. A. Sloane, Dec 05 2008

Programs

Magma

a := function(n)
if n le 2 then return n-1; end if;
if n mod 4 eq 2 then n := n div 2; end if;
f := Factorization(n);
return &+[(t[1]-1)*t[1]^(t[2]-1):t in f];
end function;

Mathematica

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 *)

Formula

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

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