A319664 - OEIS

%I #11 Oct 01 2018 21:13:42

%S 1,1,5,1,13,9,5,1,29,9,5,17,13,25,21,1,61,9,37,17,13,25,53,33,29,41,5,

%T 49,45,57,21,1,125,9,101,81,13,89,117,33,29,41,5,113,45,121,21,65,61,

%U 73,37,17,77,25,53,97,93,105,69,49,109,57,85

%N Irregular triangle read by rows: T(n,k) = (-3)^k mod 2^n, n >= 2, 0 <= k <= 2^(n-2) - 1.

%C The n-th row contains 2^(n-2) numbers, and is a permutation of 1, 5, 9, ..., 2^n - 3.

%C 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) - 1, 2) = v(k, 2) + 2, where v(k, 2) = A007814(k) is the 2-adic valuation of k. Also, T(n,k) 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.

%e Table begins

%e 1,

%e 1, 5,

%e 1, 13, 9, 5,

%e 1, 29, 9, 5, 17, 13, 25, 21,

%e 1, 61, 9, 37, 17, 13, 25, 53, 33, 29, 41, 5, 49, 45, 57, 21,

%e 1, 125, 9, 101, 81, 13, 89, 117, 33, 29, 41, 5, 113, 45, 121, 21, 65, 61, 73, 37, 17, 77, 25, 53, 97, 93, 105, 69, 49, 109, 57, 85

%e ...

%o (PARI) T(n,k) = lift(Mod(-3,2^n)^k)

%Y Cf. A007814, A319666.

%K nonn,tabf

%O 2,3

%A _Jianing Song_, Sep 25 2018