A320561 Irregular table read by rows: T(n,k) = (2*k+1)^(2*k+1) mod 2^n, 0 <= k <= 2^(n-1) - 1.

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

Offset

1,3

Comments

The sequence {k^k mod 2^n} has period 2^n. The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.

Note that the first 5 rows are the same as those in A320562, but after that they differ.

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. [Revised by Jianing Song, Nov 24 2018]

For n >= 3, T(n,k) = 2*k + 1 iff k is divisible by 2^floor((n-1)/2) or k = 2^(n-2) - 1 or k = 2^(n-1) - 1.

T(n,k) is the multiplicative inverse of A321901(n,k) modulo 2^n. - Jianing Song, Nov 24 2018

Links

Muniru A Asiru, Table of n, a(n) for n = 1..8191 (Rows n=1..13)

Formula

T(n,k) = A320602(2^n, 2*k+1).

T(n,k) = 2^n - A321901(n,2^(n-1)-1-k). - Jianing Song, Nov 24 2018

Example

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

Mathematica

Table[PowerMod[(2 k + 1), (2 k + 1), 2^n], {n, 6}, {k, 0, 2^(n - 1) - 1}] // Flatten (* Michael De Vlieger, Oct 22 2018 *)

Prog

(PARI) T(n, k) = lift(Mod(2*k+1, 2^n)^(2*k+1))
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print);
(GAP) T:= Flat(List([1..6], n->List([0..2^(n-1)-1], k->PowerMod(2*k+1, 2*k+1, 2^n)))); # Muniru A Asiru, Oct 23 2018

Crossrefs

Cf. A007814.

{x^x} and its inverse: this sequence & A320562.

{x^(-x)} and its inverse: A321901 & A321904.

{x^(1/x)} and its inverse: A321902 & A321905.

{x^(-1/x)} and its inverse: A321903 & A321906.

Sequence in context: A321903 A189442 A320562 * A321902 A321905 A323495

Adjacent sequences: A320558 A320559 A320560 * A320562 A320563 A320564

Keyword

nonn,tabf

Author

Jianing Song, Oct 15 2018

Status

approved