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A323556
Irregular table read by rows: T(n,k) = (2*k+1)^(1/3) 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, 27, 29, 23, 25, 19, 21, 15, 17, 11, 13, 7, 9, 3, 5, 31, 1, 59, 29, 23, 25, 19, 53, 47, 49, 43, 13, 7, 9, 3, 37, 31, 33, 27, 61, 55, 57, 51, 21, 15, 17, 11, 45, 39, 41, 35, 5, 63, 1, 123, 93, 23, 25, 83, 53, 47, 49, 43, 13, 71, 73, 3, 101, 95, 97, 91, 61, 119, 121, 51, 21, 15, 17, 11, 109, 39, 41, 99, 69, 63
OFFSET
1,3
COMMENTS
T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that x^3 == 2*k + 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.
T(n,k) is the multiplicative inverse of A323553(n,k) modulo 2^n.
EXAMPLE
Table starts
1,
1, 3,
1, 3, 5, 7,
1, 11, 13, 7, 9, 3, 5, 15,
1, 27, 29, 23, 25, 19, 21, 15, 17, 11, 13, 7, 9, 3, 5, 31,
1, 59, 29, 23, 25, 19, 53, 47, 49, 43, 13, 7, 9, 3, 37, 31, 33, 27, 61, 55, 57, 51, 21, 15, 17, 11, 45, 39, 41, 35, 5, 63,
...
PROG
(PARI) T(n, k) = if(n==2, 2*k+1, lift(sqrtn(2*k+1+O(2^n), 3)))
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)
CROSSREFS
Cf. also A007814.
{(2*k+1)^e mod 2^n}: A323495 (e=-1), A323553 (e=-1/3), A323554 (e=-1/5), A323555 (e=1/5), this sequence (e=1/3).
Sequence in context: A321905 A323495 A323554 * A338329 A234587 A339413
KEYWORD
nonn,tabf
AUTHOR
Jianing Song, Aug 30 2019
STATUS
approved