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A320561 Irregular table read by rows: T(n,k) = (2*k+1)^(2*k+1) mod 2^n, 0 <= k <= 2^(n-1) - 1.

%I A320561 #26 Dec 22 2018 16:48:11
%S A320561 1,1,3,1,3,5,7,1,11,5,7,9,3,13,15,1,27,21,23,9,19,29,15,17,11,5,7,25,
%T A320561 3,13,31,1,27,53,55,9,19,61,47,17,11,5,39,25,3,13,31,33,59,21,23,41,
%U A320561 51,29,15,49,43,37,7,57,35,45,63
%N A320561 Irregular table read by rows: T(n,k) = (2*k+1)^(2*k+1) mod 2^n, 0 <= k <= 2^(n-1) - 1.
%C A320561 The sequence {k^k mod 2^n} has period 2^n. The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.
%C A320561 Note that the first 5 rows are the same as those in A320562, but after that they differ.
%C A320561 For all n, k we have v(T(n,k)-1, 2) = v(k, 2) + 1 and v(T(n,k)+1, 2) = v(k+1, 2) + 1, where v(k, 2) = A007814(k) is the 2-adic valuation of k. [Revised by _Jianing Song_, Nov 24 2018]
%C A320561 For n >= 3, T(n,k) = 2*k + 1 iff k is divisible by 2^floor((n-1)/2) or k = 2^(n-2) - 1 or k = 2^(n-1) - 1.
%C A320561 T(n,k) is the multiplicative inverse of A321901(n,k) modulo 2^n. - _Jianing Song_, Nov 24 2018
%H A320561 Muniru A Asiru, <a href="/A320561/b320561.txt">Table of n, a(n) for n = 1..8191</a> (Rows n=1..13)
%F A320561 T(n,k) = A320602(2^n, 2*k+1).
%F A320561 T(n,k) = 2^n - A321901(n,2^(n-1)-1-k). - _Jianing Song_, Nov 24 2018
%e A320561 Table starts
%e A320561 1,
%e A320561 1, 3,
%e A320561 1, 3, 5, 7,
%e A320561 1, 11, 5, 7, 9, 3, 13, 15,
%e A320561 1, 27, 21, 23, 9, 19, 29, 15, 17, 11, 5, 7, 25, 3, 13, 31,
%e A320561 1, 27, 53, 55, 9, 19, 61, 47, 17, 11, 5, 39, 25, 3, 13, 31, 33, 59, 21, 23, 41, 51, 29, 15, 49, 43, 37, 7, 57, 35, 45, 63,
%e A320561 ...
%t A320561 Table[PowerMod[(2 k + 1), (2 k + 1), 2^n], {n, 6}, {k, 0, 2^(n - 1) - 1}] // Flatten (* _Michael De Vlieger_, Oct 22 2018 *)
%o A320561 (PARI) T(n,k) = lift(Mod(2*k+1, 2^n)^(2*k+1))
%o A320561 tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print);
%o A320561 (GAP) T:= Flat(List([1..6],n->List([0..2^(n-1)-1],k->PowerMod(2*k+1,2*k+1,2^n)))); # _Muniru A Asiru_, Oct 23 2018
%Y A320561 Cf. A007814.
%Y A320561 {x^x} and its inverse: this sequence & A320562.
%Y A320561 {x^(-x)} and its inverse: A321901 & A321904.
%Y A320561 {x^(1/x)} and its inverse: A321902 & A321905.
%Y A320561 {x^(-1/x)} and its inverse: A321903 & A321906.
%K A320561 nonn,tabf
%O A320561 1,3
%A A320561 _Jianing Song_, Oct 15 2018