A321906 Irregular table read by rows: T(n,k) is the smallest m such that m^(-1/m) == 2*k + 1 (mod 2^n), 0 <= k <= 2^(n-1) - 1.

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, 11, 5, 31, 1, 19, 61, 7, 57, 27, 53, 15, 49, 35, 45, 23, 41, 43, 37, 31, 33, 51, 29, 39, 25, 59, 21, 47, 17, 3, 13, 55, 9, 11, 5, 63

Offset

1,3

Comments

T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that (2*k+1)^(-m) == m (mod 2^n).

The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.

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.

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).

T(n,k) is the multiplicative inverse of A320562(n,k) modulo 2^n.

Links

Table of n, a(n) for n=1..63.

Formula

T(n,k) = 2^n - A321905(n,2^(n-1)-1-k).

Example

Table starts
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, 11, 5, 31,
1, 19, 61, 7, 57, 27, 53, 15, 49, 35, 45, 23, 41, 43, 37, 31, 33, 51, 29, 39, 25, 59, 21, 47, 17, 3, 13, 55, 9, 11, 5, 63,
...

Prog

(PARI) T(n, k) = my(m=1); while(Mod(2*k+1, 2^n)^(-m)!=m, m+=2); m
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)

Crossrefs

Cf. A007814.

{x^x} and its inverse: A320561 & A320562.

{x^(-x)} and its inverse: A321901 & A321904.

{x^(1/x)} and its inverse: A321902 & A321905.

{x^(-1/x)} and its inverse: A321903 & this sequence.

Sequence in context: A323553 A170898 A321901 * A321904 A321903 A189442

Adjacent sequences: A321903 A321904 A321905 * A321907 A321908 A321909

Keyword

nonn,tabf

Author

Jianing Song, Nov 21 2018

Status

approved