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

1,3

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.

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.

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 *)

Prog

(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;

Crossrefs

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

Sequence in context: A306275 A322321 A080737 * A293484 A000010 A003978

Adjacent sequences: A152452 A152453 A152454 * A152456 A152457 A152458

Keyword

easy,nonn

Author

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

Status

approved