OFFSET
2,5
COMMENTS
Also T(n,k) is the remainder of the 2-adic integer log_(-3)(4*k + 1) modulo 2^(n-2), since log_(-3)() is a group isomorphism from (1+4Z/2^nZ, *) to (Z/2^(n-2)Z, +). - Jianing Song, Mar 15 2026
The n-th row contains 2^(n-2) 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^(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.
LINKS
Jianing Song, Table of n, a(n) for n = 2..16384 (Rows n = 2..15; row n appears as indices 2^(n-2)+1..2^(n-1))
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
(PARI) row(n) = my(v=vector(2^(n-2))); for(i=0, 2^(n-2)-1, v[(lift(Mod(-3, 2^n)^i)+3)>>2]=i); v \\ Jianing Song, Mar 15 2026; assuming the first entries have indices 0
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Jianing Song, Sep 25 2018
EXTENSIONS
Definition clarified by Jianing Song, Mar 15 2026
STATUS
approved