A321903 Irregular table read by rows: T(n,k) = (2*k+1)^(-1/(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, 51, 61, 7, 57, 59, 53, 15, 49, 3, 45, 23, 41, 11, 37, 31, 33, 19, 29, 39, 25, 27, 21, 47, 17, 35, 13, 55, 9, 43, 5, 63

Offset

1,3

Comments

T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that x^(-(2*k+1)) == 2*k + 1 (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 A321902(n,k) modulo 2^n.

Links

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

Formula

T(n,k) = 2^n - A321902(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, 51, 61, 7, 57, 59, 53, 15, 49, 3, 45, 23, 41, 11, 37, 31, 33, 19, 29, 39, 25, 27, 21, 47, 17, 35, 13, 55, 9, 43, 5, 63,
...

Prog

(PARI) T(n, k) = my(m=1); while(Mod(m, 2^n)^(-(2*k+1))!=2*k+1, 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: this sequence & A321906.

Sequence in context: A321901 A321906 A321904 * A189442 A320562 A320561

Adjacent sequences: A321900 A321901 A321902 * A321904 A321905 A321906

Keyword

nonn,tabf

Author

Jianing Song, Nov 21 2018

Status

approved