A323554 Irregular table read by rows: T(n,k) = (2*k+1)^(-1/5) mod 2^n, 0 <= k <= 2^(n-1) - 1.

1, 1, 3, 1, 3, 5, 7, 1, 11, 13, 7, 9, 3, 5, 15, 1, 27, 29, 23, 25, 19, 21, 15, 17, 11, 13, 7, 9, 3, 5, 31, 1, 27, 61, 23, 25, 51, 21, 47, 49, 11, 45, 7, 9, 35, 5, 31, 33, 59, 29, 55, 57, 19, 53, 15, 17, 43, 13, 39, 41, 3, 37, 63, 1, 91, 61, 87, 89, 115, 85, 47, 49, 11, 109, 7, 9, 35, 5, 95, 97, 59, 29, 55, 57, 83, 53, 15, 17, 107, 77, 103, 105, 3, 101, 63

Offset

1,3

Comments

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

T(n,k) is the multiplicative inverse of A323555(n,k) modulo 2^n.

Links

Robert Israel, Table of n, a(n) for n = 1..16383 (rows 1 to 14, flattened)

Example

Table starts
1,
1, 3,
1, 3, 5, 7,
1, 11, 13, 7, 9, 3, 5, 15,
1, 27, 29, 23, 25, 19, 21, 15, 17, 11, 13, 7, 9, 3, 5, 31
1, 27, 61, 23, 25, 51, 21, 47, 49, 11, 45, 7, 9, 35, 5, 31, 33, 59, 29, 55, 57, 19, 53, 15, 17, 43, 13, 39, 41, 3, 37, 63
...- corrected by Robert Israel, Dec 15 2020

Maple

for n from 1 to 8 do
seq(numtheory:-mroot(2*k+1, -5, 2^n), k=0..2^(n-1)-1)
od; # Robert Israel, Dec 15 2020

Crossrefs

Cf. A007814.

{(2*k+1)^e mod 2^n}: A323495 (e=-1), A323553 (e=-1/3), this sequence (e=-1/5), A323555 (e=1/5), A323556 (e=1/3).

Sequence in context: A321902 A321905 A323495 * A323556 A338329 A234587

Adjacent sequences: A323551 A323552 A323553 * A323555 A323556 A323557

Keyword

nonn,tabf

Author

Jianing Song, Aug 30 2019

Extensions

Corrected by Robert Israel, Dec 15 2020

Status

approved