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A057764
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Triangle T(n,k) = number of nonzero elements of multiplicative order k in Galois field GF(2^n) (n >= 1, 1 <= k <= 2^n-1).
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3
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1, 1, 0, 2, 1, 0, 0, 0, 0, 0, 6, 1, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 1, 0, 2, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36
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OFFSET
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1,4
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LINKS
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FORMULA
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T(n,k) = A000010(k) if k is a divisor of 2^n-1, otherwise 0.
Sum_{k=1..2^n-1} T(n,k) = 2^n-1 = A000225(n).
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EXAMPLE
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Table begins:
1;
1, 0, 2;
1, 0, 0, 0, 0, 0, 6;
...
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MAPLE
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f:= proc(n, k) if 2^n-1 mod k = 0 then numtheory:-phi(k) else 0 fi end proc:
seq(seq(f(n, k), k=1..2^n-1), n=1..10); # Robert Israel, Jul 21 2016
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MATHEMATICA
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T[n_, k_] := If[Divisible[2^n - 1, k], EulerPhi[k], 0];
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PROG
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(Magma) {* Order(g) : g in GF(2^6) | g ne 0 *};
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CROSSREFS
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KEYWORD
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nonn,easy,nice,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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