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
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.
Crossrefs
Programs
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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
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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 *) -
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
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).
Extensions
More terms from Klaus Brockhaus, Mar 10 2003