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A323495
Irregular table read by rows: T(n,k) = (2*k+1)^(-1) mod 2^n, 0 <= k <= 2^(n-1) - 1.
4
1, 1, 3, 1, 3, 5, 7, 1, 11, 13, 7, 9, 3, 5, 15, 1, 11, 13, 23, 25, 3, 5, 15, 17, 27, 29, 7, 9, 19, 21, 31, 1, 43, 13, 55, 57, 35, 5, 47, 49, 27, 61, 39, 41, 19, 53, 31, 33, 11, 45, 23, 25, 3, 37, 15, 17, 59, 29, 7, 9, 51, 21, 63, 1, 43, 77, 55, 57, 35, 69, 111, 113, 27, 61, 39, 41, 19, 53, 95, 97, 11, 45, 23, 25, 3, 37, 79, 81, 123, 29, 7, 9, 115, 21, 63
OFFSET
1,3
COMMENTS
T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that x*(2*k+1) == 1 (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.
FORMULA
For n >= 3, T(n,k) = (2*k+1)^(2^(n-2)-1) mod 2^n, 0 <= k <= 2^(n-1) - 1.
EXAMPLE
Table starts
1,
1, 3,
1, 3, 5, 7,
1, 11, 13, 7, 9, 3, 5, 15,
1, 11, 13, 23, 25, 3, 5, 15, 17, 27, 29, 7, 9, 19, 21, 31,
1, 43, 13, 55, 57, 35, 5, 47, 49, 27, 61, 39, 41, 19, 53, 31, 33, 11, 45, 23, 25, 3, 37, 15, 17, 59, 29, 7, 9, 51, 21, 63,
...
PROG
(PARI) T(n, k) = lift(Mod(2*k+1, 2^n)^(-1))
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)
CROSSREFS
Cf. A007814.
{(2*k+1)^e mod 2^n}: this sequence (e=-1), A323553 (e=-1/3), A323554 (e=-1/5), A323555 (e=1/5), A323556 (e=1/3).
Sequence in context: A320561 A321902 A321905 * A323554 A323556 A338329
KEYWORD
nonn,tabf
AUTHOR
Jianing Song, Aug 30 2019
STATUS
approved