

A080737


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


10



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

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), 202209.


FORMULA

For n > 2, a(2^r) = 2^(r1) 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] = p1; 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}] (* JeanFrançois 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 !! (n1)
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 A322321 * A152455 A293484 A000010 A003978
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



