A323495 Irregular table read by rows: T(n,k) = (2*k+1)^(-1) 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, 11, 13, 23, 25, 3, 5, 15, 17, 27, 29, 7, 9, 19, 21, 31, 1, 43, 13, 55, 57, 35, 5, 47, 49, 27, 61, 39, 41, 19, 53, 31, 33, 11, 45, 23, 25, 3, 37, 15, 17, 59, 29, 7, 9, 51, 21, 63, 1, 43, 77, 55, 57, 35, 69, 111, 113, 27, 61, 39, 41, 19, 53, 95, 97, 11, 45, 23, 25, 3, 37, 79, 81, 123, 29, 7, 9, 115, 21, 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) == 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.

Links

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

Formula

For n >= 3, T(n,k) = (2*k+1)^(2^(n-2)-1) mod 2^n, 0 <= k <= 2^(n-1) - 1.

Example

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

Prog

(PARI) T(n, k) = lift(Mod(2*k+1, 2^n)^(-1))
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}: this sequence (e=-1), A323553 (e=-1/3), A323554 (e=-1/5), A323555 (e=1/5), A323556 (e=1/3).

Sequence in context: A320561 A321902 A321905 * A323554 A323556 A338329

Adjacent sequences: A323492 A323493 A323494 * A323496 A323497 A323498

Keyword

nonn,tabf

Author

Jianing Song, Aug 30 2019

Status

approved