A104235 -id:A104235 - cp's OEIS frontend

A104401 a(n) = A104235(n)/4.

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78

Offset: 0

N. J. A. Sloane, Apr 18 2005

nonn

First differs from A004773(n)=floor(4n/3) at a(47)=64 vs A004773(47)=62. - M. F. Hasler, Oct 05 2014

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A004773, A104235

A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

Original entry on oeis.org

0, 3, 6, 5, 4, 15, 10, 9, 8, 11, 14, 13, 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30, 61, 44, 39, 34, 33, 32, 35, 38, 37, 36, 47, 42, 41, 40, 43, 46, 45, 60, 55, 50, 49, 48, 51, 54, 53, 52, 63, 58, 57, 56, 59, 126, 93, 76, 71, 66, 65, 64, 67, 70, 69

Offset: 0

Philippe Deléham, Feb 13 2005

All terms are distinct, but certain terms (see A102371) are missing. But see A103122.

Trajectory of 1 is 1, 3, 5, 15, 17, 19, 21, 31, 33, ..., see A103192.

			........0
........1
.......10
.......11
......100
......101
......110
......111
.....1000
.........
The upward-sloping diagonals are:
0
11
110
101
100
1111
1010
.......
giving 0, 3, 6, 5, 4, 15, 10, ...
The sequence has a natural decomposition into blocks (see the paper): 0; 3; 6, 5, 4; 15, 10, 9, 8, 11, 14, 13; 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30; 61, ...
Reading the array of binary numbers along diagonals with slope 1 gives this sequence, slope 2 gives A105085, slope 0 gives A001477 and slope -1 gives A105033.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp. Preprint versions: [pdf, ps].
Index entries for sequences related to binary expansion of n

Related sequences (1): A103542 (binary version), A102371 (complement), A103185, A103528, A103529, A103530, A103318, A034797, A103543, A103581, A103582, A103583.
Related sequences (2): A103584, A103585, A103586, A103587, A103127, A103192 (trajectory of 1), A103122, A103588, A103589, A103202 (sorted), A103205 (base 10 version).
Related sequences (3): A103747 (trajectory of 2), A103621, A103745, A103615, A103842, A103863, A104234, A104235, A103813, A105023, A105024, A105025, A105026, A105027, A105028.
Related sequences (4): A105029, A105030, A105031, A105032, A105033, A105034, A105035, A105108.
Related sequences (5): A105229, A105271, A104378, A104401, A104403, A104489, A104490, A104853, A104893, A104894, A105085.

a102370 n = a102370_list !! n
a102370_list = 0 : map (a105027 . toInteger) a062289_list
-- Reinhard Zumkeller, Jul 21 2012

A102370:=proc(n) local t1,l; t1:=n; for l from 1 to n do if n+l mod 2^l = 0 then t1:=t1+2^l; fi; od: t1; end;

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s]; Table[ f[n] + n, {n, 0, 71}] (* Robert G. Wilson v, Mar 21 2005 *)

A102370(n)=n-1+sum(k=0,ceil(log(n+1)/log(2)),if((n+k)%2^k,0,2^k)) \\ Benoit Cloitre, Mar 20 2005

{a(n) = if( n<1, 0, sum( k=0, length( binary( n)), bitand( n + k, 2^k)))} /* Michael Somos, Mar 26 2012 */

def a(n): return 0 if n<1 else sum([(n + k)&(2**k) for k in range(len(bin(n)[2:]) + 1)]) # Indranil Ghosh, May 03 2017

a(n) = n + Sum_{ k >= 1 such that n + k == 0 mod 2^k } 2^k. (Cf. A103185.) In particular, a(n) >= n. - N. J. A. Sloane, Mar 18 2005

a(n) = A105027(A062289(n)) for n > 0. - Reinhard Zumkeller, Jul 21 2012

More terms from Benoit Cloitre, Mar 20 2005

A105271 Fixed points of the permutation of the nonnegative integers defined by A105025 (i.e., n such that A105025(n) = n).

Original entry on oeis.org

0, 1, 4, 6, 17, 21, 25, 29, 1024, 1032, 1040, 1048, 1056, 1064, 1072, 1080, 1088, 1096, 1104, 1112, 1120, 1128, 1136, 1144, 1152, 1160, 1168, 1176, 1184, 1192, 1200, 1208, 1216, 1224, 1232, 1240, 1248, 1256, 1264, 1272, 1280, 1288, 1296, 1304, 1312, 1320

Offset: 1

Emeric Deutsch, Apr 16 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A105025.

Cf. A214417, A214414, A214433, A104235.

a105271 n = a105271_list !! (n-1)
a105271_list = [x | x <- [0..], a105025 x == x]
-- Reinhard Zumkeller, Jul 21 2012

More terms from John W. Layman, Jun 03 2005

Offset corrected by Reinhard Zumkeller, Jul 21 2012

A214414 Fixed points of permutations A105027 and A214417.

Original entry on oeis.org

0, 1, 5, 7, 18, 22, 26, 30, 1031, 1039, 1047, 1055, 1063, 1071, 1079, 1087, 1095, 1103, 1111, 1119, 1127, 1135, 1143, 1151, 1159, 1167, 1175, 1183, 1191, 1199, 1207, 1215, 1223, 1231, 1239, 1247, 1255, 1263, 1271, 1279, 1287, 1295, 1303, 1311, 1319, 1327

Offset: 1

Reinhard Zumkeller, Jul 21 2012

nonn

A105027(a(n)) = A214417(a(n)) = a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A105271, A214433, A104235.

a214414 n = a214414_list !! (n-1)
a214414_list = [x | x <- [0..], a105027 x == x]

A103863 Hamming distance between n and A102370(n) (in binary).

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 1, 3, 2, 4, 4, 5, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 2, 4, 3, 5, 5, 6, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 1, 3, 2, 4, 4, 5, 0, 1, 1, 2, 0, 2, 2, 3, 0

Offset: 0

Philippe Deléham, Mar 31 2005

The Hamming distance between two strings of the same length is the number of places where they differ. - Robert G. Wilson v, Apr 12 2005

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, p. 8.

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Saleem Bhatti, Channel coding; Hamming distance.
Alexander Bogomolny, Distance Between Strings.
National Institute of Standards and Technology, Hamming distance.

Cf. A102370, A103542, A104235.

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[ Log[2, n + 1] + 2]]}, While[ k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s]; hammingdistance[n_] := Count[ IntegerDigits[ BitXor[n, f[n] + n], 2], 1]; Table[ hammingdistance[n], {n, 0, 104}] (* Robert G. Wilson v, Apr 12 2005 *)

a(A104235(n)) = 0.

More terms from Robert G. Wilson v, Apr 12 2005

cp's OEIS Frontend

A104401 a(n) = A104235(n)/4.

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A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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A105271 Fixed points of the permutation of the nonnegative integers defined by A105025 (i.e., n such that A105025(n) = n).

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A214414 Fixed points of permutations A105027 and A214417.

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A103863 Hamming distance between n and A102370(n) (in binary).

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