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A306275
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Number of values 0 < k <= n for which there are no k distinct n-th roots of unity that sum to zero.
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2
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1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 2, 12, 6, 8, 8, 16, 2, 18, 4, 12, 10, 22, 2, 20, 12, 18, 6, 28, 2, 30, 16, 20, 16, 24, 2, 36, 18, 24, 4, 40, 2, 42, 10, 8, 22, 46, 2, 42, 4, 32, 12, 52, 2, 40, 6, 36, 28, 58, 2, 60, 30, 12, 32, 48, 2, 66, 16, 44, 4, 70, 2, 72
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OFFSET
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1,3
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COMMENTS
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In the first 17 terms a(n) = phi(n) except for n=12. For primes a(p) = p - 1.
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LINKS
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FORMULA
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a(n) = #{k in {1,2,...,n} | for all subsets U of {exp(2*Pi*i*m/n)|m=0,1,...,n-1} of size #U=k we have sum(U) != 0 }.
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MAPLE
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a:= proc(n) option remember; local f, b; f, b:=
map(i-> i[1], ifactors(n)[2]),
proc(m, i) option remember; m=0 or i>0 and
(b(m, i-1) or f[i]<=m and b(m-f[i], i))
end; forget(b); (t-> add(
`if`(b(j, t) and b(n-j, t), 0, 1), j=1..n))(nops(f))
end:
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MATHEMATICA
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a := Function[{n}, Count[Function[{k}, Fold[And, (#!=0)& /@ RootReduce @* Total /@ Subsets[Exp[2*Pi*I*#/n]& /@ Range[0, n-1], {k}]]] /@ Range[1, n], True] ]
(* Second program: *)
A322366[n_] := A322366[n] = Module[{f, b}, f = FactorInteger[n][[All, 1]]; b[m_, i_] := b[m, i] = m == 0 || i > 0 && (b[m, i - 1] || f[[i]] <= m && b[m - f[[i]], i]); Function[t, Sum[If[b[j, t] && b[n - j, t], 1, 0], {j, 0, n}]][Length[f]]];
a[n_] := If[n == 1, 1, 1 + n - A322366[n]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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