A321904 Irregular table read by rows: T(n,k) is the smallest m such that m^(-m) == 2*k + 1 (mod 2^n), 0 <= k <= 2^(n-1) - 1.

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

Offset: 1

Author

Jianing Song, Nov 21 2018

Keywords

• nonn
• tabf

Comments

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 == -1 (mod 2^floor((n-1)/2)) or k = 0 or k = 2^(n-2).

T(n,k) is the multiplicative inverse of A321905(n,k) modulo 2^n.

Examples

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

Crossrefs

Cf. A007814.

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

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

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

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

Programs

PARI

T(n, k) = my(m=1); while(Mod(m, 2^n)^(-m)!=2*k+1, 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 - A320562(n,2^(n-1)-1-k).