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%I #32 Apr 04 2023 10:33:40
%S 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,
%T 6,30,16,12,16,10,8,36,18,14,8,40,8,42,12,10,22,46,10,42,20,18,14,52,
%U 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
%N a(1) = a(2) = 0; for n > 2, the least dimension of a lattice possessing a symmetry of order n.
%H Reinhard Zumkeller, <a href="/A080737/b080737.txt">Table of n, a(n) for n = 1..10000</a>
%H J. Bamberg, G. Cairns and D. Kilminster, <a href="http://www.jstor.org/stable/3647934">The crystallographic restriction, permutations and Goldbach's conjecture</a>, Amer. Math. Monthly, 110 (March 2003), 202-209.
%H Savinien Kreczman, Luca Prigioniero, Eric Rowland, and Manon Stipulanti, <a href="https://orbi.uliege.be/bitstream/2268/300422/1/paper-KPRS-submission.pdf">Magic numbers in periodic sequences</a>, Univ. Liège (Belgium, 2023). See p. 7.
%F 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).
%t 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 *)
%o (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
%o (Haskell)
%o a080737 n = a080737_list !! (n-1)
%o a080737_list = 0 : (map f [2..]) where
%o f n | mod n 4 == 2 = a080737 $ div n 2
%o | otherwise = a067240 n
%o -- _Reinhard Zumkeller_, Jun 13 2012, Jun 11 2012
%Y Cf. A080736, A080738, A080739, A080740, A067240, A000010, A141809.
%Y See A152455 for another version.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_, Mar 08 2003
%E More terms from _Klaus Brockhaus_, Mar 10 2003
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