A323556 Irregular table read by rows: T(n,k) = (2*k+1)^(1/3) 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, 59, 29, 23, 25, 19, 53, 47, 49, 43, 13, 7, 9, 3, 37, 31, 33, 27, 61, 55, 57, 51, 21, 15, 17, 11, 45, 39, 41, 35, 5, 63, 1, 123, 93, 23, 25, 83, 53, 47, 49, 43, 13, 71, 73, 3, 101, 95, 97, 91, 61, 119, 121, 51, 21, 15, 17, 11, 109, 39, 41, 99, 69, 63

Offset

1,3

Comments

T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that x^3 == 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 A323553(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, 11, 13, 7, 9, 3, 5, 15,
1, 27, 29, 23, 25, 19, 21, 15, 17, 11, 13, 7, 9, 3, 5, 31,
1, 59, 29, 23, 25, 19, 53, 47, 49, 43, 13, 7, 9, 3, 37, 31, 33, 27, 61, 55, 57, 51, 21, 15, 17, 11, 45, 39, 41, 35, 5, 63,
...

Prog

(PARI) T(n, k) = if(n==2, 2*k+1, lift(sqrtn(2*k+1+O(2^n), 3)))
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)

Crossrefs

Cf. A322701, A322926, A322934, A322999.

Cf. also A007814.

{(2*k+1)^e mod 2^n}: A323495 (e=-1), A323553 (e=-1/3), A323554 (e=-1/5), A323555 (e=1/5), this sequence (e=1/3).

Sequence in context: A321905 A323495 A323554 * A338329 A234587 A339413

Adjacent sequences: A323553 A323554 A323555 * A323557 A323558 A323559

Keyword

nonn,tabf

Author

Jianing Song, Aug 30 2019

Status

approved