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A080737
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a(1) = a(2) = 0; for n > 2, the least dimension of a lattice possessing a symmetry of order n.
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10
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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
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OFFSET
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1,3
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LINKS
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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.
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FORMULA
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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).
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Mar 08 2003
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EXTENSIONS
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More terms from Klaus Brockhaus, Mar 10 2003
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STATUS
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approved
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