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%I A319664 #11 Oct 01 2018 21:13:42
%S A319664 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 A319664 49,45,57,21,1,125,9,101,81,13,89,117,33,29,41,5,113,45,121,21,65,61,
%U A319664 73,37,17,77,25,53,97,93,105,69,49,109,57,85
%N A319664 Irregular triangle read by rows: T(n,k) = (-3)^k mod 2^n, n >= 2, 0 <= k <= 2^(n-2) - 1.
%C A319664 The n-th row contains 2^(n-2) numbers, and is a permutation of 1, 5, 9, ..., 2^n - 3.
%C A319664 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 A319664 Table begins
%e A319664 1,
%e A319664 1, 5,
%e A319664 1, 13, 9, 5,
%e A319664 1, 29, 9, 5, 17, 13, 25, 21,
%e A319664 1, 61, 9, 37, 17, 13, 25, 53, 33, 29, 41, 5, 49, 45, 57, 21,
%e A319664 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 A319664 ...
%o A319664 (PARI) T(n,k) = lift(Mod(-3,2^n)^k)
%Y A319664 Cf. A007814, A319666.
%K A319664 nonn,tabf
%O A319664 2,3
%A A319664 _Jianing Song_, Sep 25 2018

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