A295653 -id:A295653 - cp's OEIS frontend

A298486 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 + m can be computed with carry in decimal base.

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 0, 1, 20, 11, 11, 11, 11, 11, 11, 11, 11, 11, 0, 1

Offset: 0

Rémy Sigrist, Jan 20 2018

The corresponding sequence for the binary base is A295653.

			Square array begins:
  n\k|  0   1   2   3   4   5   6   7   8   9   10  ...
  ---+-------------------------------------------------
    0|  0   1   2   3   4   5   6   7   8   9   10  ... <-- A001477
    1|  0   1   2   3   4   5   6   7   8  10   11  ...
    2|  0   1   2   3   4   5   6   7  10  11   12  ...
    3|  0   1   2   3   4   5   6  10  11  12   13  ...
    4|  0   1   2   3   4   5  10  11  12  13   14  ...
    5|  0   1   2   3   4  10  11  12  13  14   20  ...
    6|  0   1   2   3  10  11  12  13  20  21   22  ...
    7|  0   1   2  10  11  12  20  21  22  30   31  ...
    8|  0   1  10  11  20  21  30  31  40  41   50  ...
    9|  0  10  20  30  40  50  60  70  80  90  100  ... <-- A008592
   10|  0   1   2   3   4   5   6   7   8   9   10  ...

Rémy Sigrist, Table of n, a(n) for n = 0..5150

Cf. A001477, A004218, A008592, A122840, A295653, A298372.

T(n,k,{base=10}) = my (v=0, p=1); while (k, my (r=base - (n%base)); v += p*(k%r); n \= base; k \= r; p *= base); v

For any n >= 0 and k >= 0:
- T(0, k) = k,
- T(9, k) = 10 * k,
- T(10^n - 1, k) = 10^n * k,
- T(n, 0) = 0,
- T(n, 1) = 10^A122840(n+1),
- T(n, k + A298372(n)) = k + 10^A004218(n+1) (i.e. each row is linear).

A352993 a(n) is the n-th positive integer that has no common 1-bit with n; a(0) = 0.

Original entry on oeis.org

0, 2, 4, 12, 8, 18, 24, 56, 16, 34, 36, 84, 48, 98, 112, 240, 32, 66, 68, 140, 72, 162, 168, 360, 96, 194, 196, 420, 224, 450, 480, 992, 64, 130, 132, 268, 136, 274, 280, 600, 144, 322, 324, 660, 336, 706, 720, 1488, 192, 386, 388, 780, 392, 834, 840, 1736

Offset: 0

Rémy Sigrist, Apr 14 2022

This sequence corresponds to the main diagonal of A295653.
To compute a(n):
- consider the binary expansion of n: Sum_{k >= 0} b_k * 2^k,
- and the positions of zeros in this binary expansion: {z_k, k >= 0},
- then a(n) = Sum_{k >= 0} b_k * 2^z(k).

			For n = 43:
- the binary expansion of 43 is       "... 0 0 0 0 1 0 1 0 1 1"
- so the binary expansion of a(43) is "... 1 0 1 0(0)1(0)1(0 0)",
- and a(43) = 660.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A000120, A020522, A070939, A295653.

a(n) = { my (m=n, v=0); for (e=0, oo, if (n==0, return (v), !bittest(m, e), if (n%2, v+=2^e;); n\=2)) }

def a(n):
    b = bin(n)[2:][::-1]
    z = [k for k, bk in enumerate(b+'0'*(len(b)-b.count('0'))) if bk=='0']
    return sum(int(bk)*2**zk for bk, zk in zip(b, z))
print([a(n) for n in range(56)]) # Michael S. Branicky, Apr 21 2022

a(n) = A295653(n, n).
a(2^k) = 2^(k+1) for any k >= 0.
a(2^k-1) = A020522(k) for any k >= 0.
A000120(a(n)) = A000120(n).
A070939(a(n)) = A070939(n) + A000120(n).

cp's OEIS Frontend

A298486 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 + m can be computed with carry in decimal base.

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