A228083
Table of binary Self-numbers and their descendants; square array T(r,c), with row r>=1, column c>=1, read by antidiagonals.
Original entry on oeis.org
1, 2, 4, 3, 5, 6, 5, 7, 8, 13, 7, 10, 9, 16, 15, 10, 12, 11, 17, 19, 18, 12, 14, 14, 19, 22, 20, 21, 14, 17, 17, 22, 25, 22, 24, 23, 17, 19, 19, 25, 28, 25, 26, 27, 30, 19, 22, 22, 28, 31, 28, 29, 31, 34, 32, 22, 25, 25, 31, 36, 31, 33, 36, 36, 33, 37
Offset: 1
The top-left corner of the square array:
1, 2, 3, 5, 7, 10, 12, 14, ...
4, 5, 7, 10, 12, 14, 17, 19, ...
6, 8, 9, 11, 14, 17, 19, 22, ...
13, 16, 17, 19, 22, 25, 28, 31, ...
15, 19, 22, 25, 28, 31, 36, 38, ...
18, 20, 22, 25, 28, 31, 36, 38, ...
21, 24, 26, 29, 33, 35, 38, 41, ...
23, 27, 31, 36, 38, 41, 44, 47, ...
...
The non-initial terms on each row are obtained by adding to the preceding term the number of 1-bits in its binary representation (A000120).
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nmax0 = 100;
nmax := Length[col[1]];
col[1] = Table[n + DigitCount[n, 2, 1], {n, 0, nmax0}] // Complement[Range[Last[#]], #]&;
col[k_] := col[k] = col[k - 1] + DigitCount[col[k-1], 2, 1];
T[n_, k_] := col[k][[n]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 28 2020 *)
A230058
Numbers of the form k + wt(k) for at least two distinct k, where wt(k) = A000120(k) is the binary weight of k.
Original entry on oeis.org
5, 14, 17, 19, 22, 31, 33, 36, 38, 47, 50, 52, 55, 64, 67, 70, 79, 82, 84, 87, 96, 98, 101, 103, 112, 115, 117, 120, 129, 131, 132, 134, 143, 146, 148, 151, 160, 162, 165, 167, 176, 179, 181, 184, 193, 196, 199, 208, 211, 213, 216, 225, 227, 230, 232, 241, 244
Offset: 1
5 = 3 + 2 = 4 + 1, so 5 is in this list.
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Sort[Transpose[Select[Tally[Table[k + Total[IntegerDigits[k, 2]], {k, 0, 300}]], #[[2]] > 1 &]][[1]]] (* T. D. Noe, Oct 09 2013 *)
A228090
Numbers k for which a sum k + bitcount(k) cannot be obtained as a sum k2 + bitcount(k2) for any other k2<>k . Here bitcount(k) (A000120) gives the number of 1's in binary representation of nonnegative integer k.
Original entry on oeis.org
0, 1, 2, 5, 6, 7, 8, 9, 10, 13, 18, 21, 22, 23, 24, 25, 26, 30, 33, 37, 38, 39, 40, 41, 42, 45, 50, 53, 54, 55, 56, 57, 58, 61, 63, 64, 66, 69, 70, 71, 72, 73, 74, 77, 82, 85, 86, 87, 88, 89, 90, 94, 97, 101, 102, 103, 104, 105, 106, 109, 114, 117, 118, 119, 120
Offset: 1
0 is in this sequence because the sum 0+A000120(0)=0 cannot be obtained with any other value of k than k=0.
1 is in this sequence because the sum 1+A000120(1)=2 cannot be obtained with any other value of k than k=1.
2 is in this sequence because the sum 2+A000120(2)=3 cannot be obtained with any other value of k than k=2.
3 is not in this sequence because the sum 3+A000120(3)=5 can also be obtained with value k=4, as also 4+A000120(4)=5.
A228089
Integers k for which a sum k + bitcount(k) cannot be obtained as a sum k2 + bitcount(k2) for any other k2<>k.
Original entry on oeis.org
0, 1, 2, 5, 6, 8, 7, 9, 10, 13, 18, 21, 22, 24, 23, 25, 26, 30, 33, 37, 38, 40, 39, 41, 42, 45, 50, 53, 54, 56, 55, 57, 58, 64, 61, 66, 63, 69, 70, 72, 71, 73, 74, 77, 82, 85, 86, 88, 87, 89, 90, 94, 97, 101, 102, 104, 103, 105, 106, 109, 114, 117, 118, 120, 119
Offset: 1
6 is in this sequence because the sum 6+A000120(6)=8 cannot be obtained with any other value of k than k=6.
8 is in this sequence because the sum 8+A000120(8)=9 cannot be obtained with any other value of k than k=8.
7 is in this sequence because the sum 7+A000120(7)=10 cannot be obtained with any other value of k than k=7.
In this sequence 8 becomes before 7 because 8+A000120(8) < 7+A000120(7).
A228090 gives the same terms sorted into ascending order.
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