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%I #21 Dec 22 2018 16:48:22
%S 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 3,13,31,1,27,21,55,9,19,29,47,17,11,37,39,25,3,45,31,33,59,53,23,41,
%U 51,61,15,49,43,5,7,57,35,13,63
%N Irregular table read by rows: T(n,k) is the smallest m such that m^m == 2*k + 1 (mod 2^n), 0 <= k <= 2^(n-1) - 1.
%C 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 Note that the first 5 rows are the same as those in A320561, but after that they differ.
%C 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 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 T(n,k) is the multiplicative inverse of A321906(n,k) modulo 2^n. - _Jianing Song_, Nov 24 2018
%H Jianing Song, <a href="/A320562/b320562.txt">Table of n, a(n) for n = 1..8191</a> (Rows n=1..13)
%F For given n >= 2 and 0 <= k <= 2^(n-2) - 1, T(n,k) = T(n-1,k) if T(n-1,k)^T(n-1,k) == 2*k + 1 (mod 2^n), otherwise T(n-1,k) + 2^(n-1); for 2^(n-2) <= k <= 2^(n-1) - 1, T(n,k) = T(n,k-2^(n-2)) + 2^(n-1) if T(n,k) < 2^(n-1), otherwise T(n,k-2^(n-2)) - 2^(n-1).
%F T(n,k) = 2^n - A321904(n,2^(n-1)-1-k). - _Jianing Song_, Nov 24 2018
%e Table starts
%e 1,
%e 1, 3,
%e 1, 3, 5, 7,
%e 1, 11, 5, 7, 9, 3, 13, 15,
%e 1, 27, 21, 23, 9, 19, 29, 15, 17, 11, 5, 7, 25, 3, 13, 31,
%e 1, 27, 21, 55, 9, 19, 29, 47, 17, 11, 37, 39, 25, 3, 45, 31, 33, 59, 53, 23, 41, 51, 61, 15, 49, 43, 5, 7, 57, 35, 13, 63,
%e ...
%t Table[Block[{m = 1}, While[PowerMod[m, m, 2^n] != Mod[2 k + 1, 2^n], m++]; m], {n, 6}, {k, 0, 2^(n - 1) - 1}] // Flatten (* _Michael De Vlieger_, Oct 22 2018 *)
%o (PARI) T(n,k) = my(m=1); while(Mod(m, 2^n)^m!=2*k+1, m+=2); m
%o tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print);
%Y Cf. A007814.
%Y {x^x} and its inverse: A320561 & this sequence.
%Y {x^(-x)} and its inverse: A321901 & A321904.
%Y {x^(1/x)} and its inverse: A321902 & A321905.
%Y {x^(-1/x)} and its inverse: A321903 & A321906.
%K nonn,tabf
%O 1,3
%A _Jianing Song_, Oct 15 2018
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