

A319666


Irregular triangle read by rows: T(n,k) = log_(3)(4*k + 1) mod 2^n, n >= 2, 0 <= k <= 2^(n2)  1.


1



0, 0, 1, 0, 3, 2, 1, 0, 3, 2, 5, 4, 7, 6, 1, 0, 11, 2, 5, 4, 15, 6, 9, 8, 3, 10, 13, 12, 7, 14, 1, 0, 11, 2, 5, 20, 15, 22, 9, 8, 19, 10, 13, 28, 23, 30, 17, 16, 27, 18, 21, 4, 31, 6, 25, 24, 3, 26, 29, 12, 7, 14, 1, 0, 11, 2, 5, 52, 47, 22, 9, 40, 19, 42, 13, 28, 55, 62, 17, 16, 27, 18, 21, 4, 63, 38, 25, 56, 35, 58, 29, 44, 7, 14, 33
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OFFSET

2,5


COMMENTS

The nth row contains 2^(n2) numbers. T(n,k) is the smallest e such that (3)^e == 4*k + 1 (mod 2^n). This equation always has a solution in [0, 2^(n2)  1], so the nth row is a permutation of 0, 1, 2, ..., 2^(n2)  1.
For e >= 4, the multiplicative order of a modulo 2^e equals to 2^(e2) iff a == 3, 5 (mod 8); for e >= 5, the multiplicative order of a modulo 2^e equals to 2^(e3) iff a == 7, 9 (mod 16); for e >= 6, the multiplicative order of a modulo 2^e equals to 2^(e4) 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 2adic 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^(n3)) for 0 <= k <= 2^(n2)  1 (i denotes the imaginary unit), Chi(n,4*k+3) = Chi(n,2^n4*k3) for 0 <= k <= 2^(n2)  1, Chi(n,2^n+k) = Chi(n,k) for all integers k, then Chi(n,k) forms a Dirichlet character modulo 2^n.


LINKS

Table of n, a(n) for n=2..96.


EXAMPLE

Table begins
0,
0, 1,
0, 3, 2, 1,
0, 3, 2, 5, 4, 7, 6, 1,
0, 11, 2, 5, 4, 15, 6, 9, 8, 3, 10, 13, 12, 7, 14, 1,
0, 11, 2, 5, 20, 15, 22, 9, 8, 19, 10, 13, 28, 23, 30, 17, 16, 27, 18, 21, 4, 31, 6, 25, 24, 3, 26, 29, 12, 7, 14, 1,
...


PROG

(PARI) T(n, k) = my(i=0); while(Mod(3, 2^n)^i!=4*k+1, i++); i


CROSSREFS

Cf. A007814, A319664.
Sequence in context: A215486 A083721 A158459 * A307333 A031251 A194885
Adjacent sequences: A319663 A319664 A319665 * A319667 A319668 A319669


KEYWORD

nonn,tabf


AUTHOR

Jianing Song, Sep 25 2018


STATUS

approved



