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URL: https://oeis.org/A295653

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A295653
Square array T(n, k), n >= 0, k >= 0, read by antidiagonals upwards: T(n, k) = the (k+1)-th nonnegative number m such that n AND m = 0 (where AND denotes the bitwise AND operator).
2
0, 0, 1, 0, 2, 2, 0, 1, 4, 3, 0, 4, 4, 6, 4, 0, 1, 8, 5, 8, 5, 0, 2, 2, 12, 8, 10, 6, 0, 1, 8, 3, 16, 9, 12, 7, 0, 8, 8, 10, 8, 20, 12, 14, 8, 0, 1, 16, 9, 16, 9, 24, 13, 16, 9, 0, 2, 2, 24, 16, 18, 10, 28, 16, 18, 10, 0, 1, 4, 3, 32, 17, 24, 11, 32, 17, 20
OFFSET
0,5
COMMENTS
This sequence has similarities with A126572: here we check for common bits in binary representations, there for common primes in prime factorizations.
For any n >= 0 and k >= 0:
- T(0, k) = k,
- T(1, k) = 2*k,
- T(2, k) = A042948(k),
- T(3, k) = 4*k,
- T(4, k) = A047476(k),
- T(5, k) = A047467(k),
- T(2^n - 1, k) = 2^n * k,
- T(n, 0) = 0,
- T(n, 1) = A006519(n+1),
- T(n, k + 2^A080791(n)) = T(n, k) + 2^A029837(n+1) (i.e. each row is linear),
- A000120(T(n, k)) = A000120(k).
FORMULA
For any n >= 0 and k >= 0:
- T(0, k) = k,
- T(2*n + 1, k) = 2*T(n, k),
- T(2*n, 2*k) = 2*T(n, k),
- T(2*n, 2*k + 1) = 2*T(n, k) + 1.
For any n >= 0, T(n, k) ~ 2^A000120(n) * k as k tends to infinity.
EXAMPLE
Square array begins:
n\k 0 1 2 3 4 5 6 7 8 9 ...
0: 0 1 2 3 4 5 6 7 8 9 ...
1: 0 2 4 6 8 10 12 14 16 18 ...
2: 0 1 4 5 8 9 12 13 16 17 ...
3: 0 4 8 12 16 20 24 28 32 36 ...
4: 0 1 2 3 8 9 10 11 16 17 ...
5: 0 2 8 10 16 18 24 26 32 34 ...
6: 0 1 8 9 16 17 24 25 32 33 ...
7: 0 8 16 24 32 40 48 56 64 72 ...
8: 0 1 2 3 4 5 6 7 16 17 ...
9: 0 2 4 6 16 18 20 22 32 34 ...
PROG
(PARI) T(n, k) = if (n==0, k, n%2, 2*T(n\2, k), 2*T(n\2, k\2) + (k%2))
KEYWORD
nonn,tabl,base
AUTHOR
Rémy Sigrist, Nov 25 2017
STATUS
approved