A321905 Irregular table read by rows: T(n,k) is the smallest m such that m^(1/m) == 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, 59, 53, 55, 9, 51, 61, 47, 17, 43, 5, 39, 25, 35, 13, 31, 33, 27, 21, 23, 41, 19, 29, 15, 49, 11, 37, 7, 57, 3, 45, 63

Offset

1,3

Comments

T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that (2*k+1)^m == m (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.

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 A321904(n,k) modulo 2^n.

Links

Table of n, a(n) for n=1..63.

Formula

T(n,k) = 2^n - A321906(n,2^(n-1)-1-k).

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

Prog

(PARI) T(n, k) = my(m=1); while(Mod(2*k+1, 2^n)^m!=m, m+=2); m
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)

Crossrefs

Cf. A007814.

{x^x} and its inverse: A320561 & A320562.

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

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

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

Sequence in context: A320562 A320561 A321902 * A323495 A323554 A323556

Adjacent sequences: A321902 A321903 A321904 * A321906 A321907 A321908

Keyword

nonn,tabf

Author

Jianing Song, Nov 21 2018

Status

approved