cp's OEIS Frontend

A320562 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 A320562 #21 Dec 22 2018 16:48:22
%S A320562 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 A320562 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 A320562 51,61,15,49,43,5,7,57,35,13,63
%N A320562 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 A320562 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 A320562 Note that the first 5 rows are the same as those in A320561, but after that they differ.
%C A320562 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 A320562 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 A320562 T(n,k) is the multiplicative inverse of A321906(n,k) modulo 2^n. - _Jianing Song_, Nov 24 2018
%H A320562 Jianing Song, <a href="/A320562/b320562.txt">Table of n, a(n) for n = 1..8191</a> (Rows n=1..13)
%F A320562 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 A320562 T(n,k) = 2^n - A321904(n,2^(n-1)-1-k). - _Jianing Song_, Nov 24 2018
%e A320562 Table starts
%e A320562 1,
%e A320562 1, 3,
%e A320562 1, 3, 5, 7,
%e A320562 1, 11, 5, 7, 9, 3, 13, 15,
%e A320562 1, 27, 21, 23, 9, 19, 29, 15, 17, 11, 5, 7, 25, 3, 13, 31,
%e A320562 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 A320562 ...
%t A320562 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 A320562 (PARI) T(n,k) = my(m=1); while(Mod(m, 2^n)^m!=2*k+1, m+=2); m
%o A320562 tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print);
%Y A320562 Cf. A007814.
%Y A320562 {x^x} and its inverse: A320561 & this sequence.
%Y A320562 {x^(-x)} and its inverse: A321901 & A321904.
%Y A320562 {x^(1/x)} and its inverse: A321902 & A321905.
%Y A320562 {x^(-1/x)} and its inverse: A321903 & A321906.
%K A320562 nonn,tabf
%O A320562 1,3
%A A320562 _Jianing Song_, Oct 15 2018