%I #12 Oct 06 2018 08:47:40
%S 1,1,5,1,5,9,13,1,5,25,29,17,21,9,13,1,5,25,61,49,53,9,45,33,37,57,29,
%T 17,21,41,13,1,5,25,125,113,53,9,45,97,101,121,93,81,21,105,13,65,69,
%U 89,61,49,117,73,109,33,37,57,29,17,85,41,77
%N Irregular triangle read by rows: T(n,k) = 5^k mod 2^n, n >= 2, 0 <= k <= 2^(n2)  1.
%C The nth row contains 2^(n2) 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^(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)  1, 2) = v(k, 2) + 2, where v(k, 2) = A007814(k) is the 2adic 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, 5, 9, 13,
%e 1, 5, 25, 29, 17, 21, 9, 13,
%e 1, 5, 25, 61, 49, 53, 9, 45, 33, 37, 57, 29, 17, 21, 41, 13,
%e 1, 5, 25, 125, 113, 53, 9, 45, 97, 101, 121, 93, 81, 21, 105, 13, 65, 69, 89, 61, 49, 117, 73, 109, 33, 37, 57, 29, 17, 85, 41, 77
%e ...
%o (PARI) T(n,k) = lift(Mod(5,2^n)^k)
%Y Cf. A007814, A319665.
%K nonn,tabf
%O 2,3
%A _Jianing Song_, Sep 25 2018
