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).

Keyword

nonn,tabf

Author

Jianing Song, Aug 30 2019

Extensions

Corrected by Robert Israel, Dec 15 2020

Status

approved

A323555 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, 3, 5, 7, 9, 11, 13, 15, 1, 19, 21, 7, 9, 27, 29, 15, 17, 3, 5, 23, 25, 11, 13, 31, 1, 19, 21, 39, 41, 59, 61, 15, 17, 35, 37, 55, 57, 11, 13, 31, 33, 51, 53, 7, 9, 27, 29, 47, 49, 3, 5, 23, 25, 43, 45, 63, 1, 83, 21, 103, 105, 59, 125, 79, 81, 35, 101, 55, 57, 11, 77, 31, 33, 115, 53, 7, 9, 91, 29, 111, 113, 67, 5, 87, 89, 43, 109, 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 (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 A323554(n,k) modulo 2^n.

Links

Table of n, a(n) for n=1..95.

Example

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

Prog

(PARI) T(n, k) = if(n==2, 2*k+1, lift(sqrtn(2*k+1+O(2^n), 5)))
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)

Crossrefs

Cf. A007814.

{(2*k+1)^e mod 2^n}: A323495 (e=-1), A323553 (e=-1/3), A323554 (e=-1/5), this sequence (e=1/5), A323556 (e=1/3).

Keyword

nonn,tabf

Author

Jianing Song, Aug 30 2019

Status

approved