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

Author

Jianing Song, Nov 21 2018

Keywords

• nonn
• tabf

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.

Examples

			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,
...
		

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.

Programs

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)

Formula

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