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A321901
Irregular table read by rows: T(n,k) = (2*k+1)^(-(2*k+1)) mod 2^n, 0 <= k <= 2^(n-1) - 1.
7
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, 19, 29, 7, 57, 27, 21, 15, 49, 35, 13, 23, 41, 43, 5, 31, 33, 51, 61, 39, 25, 59, 53, 47, 17, 3, 45, 55, 9, 11, 37, 63
OFFSET
1,3
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 A320561(n,k) modulo 2^n.
FORMULA
T(n,k) = 2^n - A320561(n,2^(n-1)-1-k).
EXAMPLE
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, 19, 29, 7, 57, 27, 21, 15, 49, 35, 13, 23, 41, 43, 5, 31, 33, 51, 61, 39, 25, 59, 53, 47, 17, 3, 45, 55, 9, 11, 37, 63,
...
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)
CROSSREFS
Cf. A007814.
{x^x} and its inverse: A320561 & A320562.
{x^(-x)} and its inverse: this sequence & A321904.
{x^(1/x)} and its inverse: A321902 & A321905.
{x^(-1/x)} and its inverse: A321903 & A321906.
Sequence in context: A323555 A323553 A170898 * A321906 A321904 A321903
KEYWORD
nonn,tabf
AUTHOR
Jianing Song, Nov 21 2018
STATUS
approved