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A152455
a(n) = minimal integer m such that there exists an m X m integer matrix of order n.
4
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
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 A372681
KEYWORD
easy,nonn
AUTHOR
W. R. Unger (billu(AT)maths.usyd.edu.au), Dec 04 2008
STATUS
approved