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A321904 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.

%I A321904 #14 Feb 09 2020 02:42:47
%S A321904 1,1,3,1,3,5,7,1,3,13,7,9,11,5,15,1,19,29,7,25,27,21,15,17,3,13,23,9,
%T A321904 11,5,31,1,51,29,7,57,59,21,15,49,3,13,23,41,11,5,31,33,19,61,39,25,
%U A321904 27,53,47,17,35,45,55,9,43,37,63
%N A321904 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 A321904 The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.
%C A321904 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.
%C A321904 For n >= 3, T(n,k) = 2*k + 1 iff k == -1 (mod 2^floor((n-1)/2)) or k = 0 or k = 2^(n-2).
%C A321904 T(n,k) is the multiplicative inverse of A321905(n,k) modulo 2^n.
%F A321904 T(n,k) = 2^n - A320562(n,2^(n-1)-1-k).
%e A321904 Table starts
%e A321904 1,
%e A321904 1, 3,
%e A321904 1, 3, 5, 7,
%e A321904 1, 3, 13, 7, 9, 11, 5, 15,
%e A321904 1, 19, 29, 7, 25, 27, 21, 15, 17, 3, 13, 23, 9, 11, 5, 31,
%e A321904 1, 51, 29, 7, 57, 59, 21, 15, 49, 3, 13, 23, 41, 11, 5, 31, 33, 19, 61, 39, 25, 27, 53, 47, 17, 35, 45, 55, 9, 43, 37, 63,
%e A321904 ...
%o A321904 (PARI) T(n, k) = my(m=1); while(Mod(m, 2^n)^(-m)!=2*k+1, m+=2); m
%o A321904 tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)
%Y A321904 Cf. A007814.
%Y A321904 {x^x} and its inverse: A320561 & A320562.
%Y A321904 {x^(-x)} and its inverse: A321901 & this sequence.
%Y A321904 {x^(1/x)} and its inverse: A321902 & A321905.
%Y A321904 {x^(-1/x)} and its inverse: A321903 & A321906.
%K A321904 nonn,tabf
%O A321904 1,3
%A A321904 _Jianing Song_, Nov 21 2018