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A319665
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Irregular triangle read by rows: T(n,k) = log_5(4*k + 1) mod 2^n, n >= 2, 0 <= k <= 2^(n-2) - 1.
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1
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0, 0, 1, 0, 1, 2, 3, 0, 1, 6, 7, 4, 5, 2, 3, 0, 1, 6, 15, 12, 13, 2, 11, 8, 9, 14, 7, 4, 5, 10, 3, 0, 1, 6, 15, 28, 13, 2, 27, 24, 25, 30, 7, 20, 5, 26, 19, 16, 17, 22, 31, 12, 29, 18, 11, 8, 9, 14, 23, 4, 21, 10, 3, 0, 1, 6, 47, 28, 45, 2, 59, 56, 25, 62, 7, 20, 5, 58, 19, 48, 49, 54, 31, 12, 29, 50, 43, 40, 9, 46, 55, 4, 53, 42, 3
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OFFSET
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2,6
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COMMENTS
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The n-th row contains 2^(n-2) numbers. T(n,k) is the smallest e such that 5^e == 4*k + 1 (mod 2^n). This equation always has a solution in [0, 2^(n-2) - 1], so the n-th row is a permutation of 0, 1, 2, ..., 2^(n-2) - 1.
For e >= 4, the multiplicative order of a modulo 2^e equals to 2^(e-2) iff a == 3, 5 (mod 8); for e >= 5, the multiplicative order of a modulo 2^e equals to 2^(e-3) iff a == 7, 9 (mod 16); for e >= 6, the multiplicative order of a modulo 2^e equals to 2^(e-4) iff a == 15, 17 (mod 32), etc. From this we can see v(T(n,k), 2) = v(k, 2), where v(k, 2) = A007814(k) is the 2-adic valuation of k. Also, 4*k + 1 is a 2^v(k, 2)-th power residue but not a 2^(v(k, 2)+1)-th power residue modulo 2^i, i >= v(k, 2) + 3.
Define Chi(n,k) as: Chi(n,2*k) = 0 for all integers k, Chi(n,4*k+1) = exp(T(n,k)*Pi*i/2^(n-3)) for 0 <= k <= 2^(n-2) - 1 (i denotes the imaginary unit), Chi(n,4*k+3) = Chi(n,2^n-4*k-3) for 0 <= k <= 2^(n-2) - 1, Chi(n,2^n+k) = Chi(n,k) for all integers k, then Chi(n,k) forms a Dirichlet character modulo 2^n.
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LINKS
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Table of n, a(n) for n=2..96.
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EXAMPLE
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Table begins
0,
0, 1,
0, 1, 2, 3,
0, 1, 6, 7, 4, 5, 2, 3,
0, 1, 6, 15, 12, 13, 2, 11, 8, 9, 14, 7, 4, 5, 10, 3,
0, 1, 6, 15, 28, 13, 2, 27, 24, 25, 30, 7, 20, 5, 26, 19, 16, 17, 22, 31, 12, 29, 18, 11, 8, 9, 14, 23, 4, 21, 10, 3,
...
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PROG
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(PARI) T(n, k) = my(i=0); while(Mod(5, 2^n)^i!=4*k+1, i++); i
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CROSSREFS
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Cf. A007814, A319663.
Sequence in context: A163465 A360380 A263230 * A004443 A171616 A323883
Adjacent sequences: A319662 A319663 A319664 * A319666 A319667 A319668
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KEYWORD
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nonn,tabf
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AUTHOR
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Jianing Song, Sep 25 2018
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STATUS
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approved
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