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).

Sequence in context: A160552 A256263 A006257 * A323553 A170898 A321901

Adjacent sequences: A323552 A323553 A323554 * A323556 A323557 A323558

Keyword

nonn,tabf

Author

Jianing Song, Aug 30 2019

Status

approved