A080737 a(1) = a(2) = 0; for n > 2, the least dimension of a lattice possessing a symmetry of order n.

0, 0, 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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Savinien Kreczman, Luca Prigioniero, Eric Rowland, and Manon Stipulanti, Magic numbers in periodic sequences, Univ. Liège (Belgium, 2023). See p. 7.

Formula

For n > 2, a(2^r) = 2^(r-1) with r>1, a(p^r) = phi(p^r) with p > 2 prime, r >= 1, where phi is Euler's function A000010; in general if a(Product p_i^e_i) = Sum a(p_i^e_i).

Mathematica

a[1] = a[2] = 0; a[p_?PrimeQ] := a[p] = p-1; a[n_] := a[n] = If[Length[fi = FactorInteger[n]] == 1, EulerPhi[n], Total[a /@ (fi[[All, 1]]^fi[[All, 2]])]]; Table[a[n], {n, 1, 78}] (* Jean-François Alcover, Jun 20 2012 *)

Prog

(PARI) for(n=1, 78, k=0; if(n>1, f=factor(n); k=sum(j=1, matsize(f)[1], eulerphi(f[j, 1]^f[j, 2])); if(f[1, 1]==2&&f[1, 2]==1, k--)); print1(k, ", ")) \\ Klaus Brockhaus, Mar 10 2003
(Haskell)
a080737 n = a080737_list !! (n-1)
a080737_list = 0 : (map f [2..]) where
f n | mod n 4 == 2 = a080737 $ div n 2
| otherwise = a067240 n
-- Reinhard Zumkeller, Jun 13 2012, Jun 11 2012

Crossrefs

Cf. A080736, A080738, A080739, A080740, A067240, A000010, A141809.

See A152455 for another version.

Sequence in context: A011773 A306275 A322321 * A152455 A293484 A000010

Adjacent sequences: A080734 A080735 A080736 * A080738 A080739 A080740

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 08 2003

Extensions

More terms from Klaus Brockhaus, Mar 10 2003

Status

approved