A103542 -id:A103542 - cp's OEIS frontend

A103615 Number of zeros in A103542(n) (binary equivalent of A102370(n)).

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

Offset: 0

Philippe Deléham, Mar 31 2005

			The sequence has a natural decomposition into blocks (see the paper): 1; 0; 1, 1, 2; 0, 2, 2, 3, 1, 1, 1; 2, 1, 3, 3, 4, 2, 2, 2, 3, 0, 2, 2, 3, 1, 1; 1, 3, ...

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

Cf. A102370, A103542, A089400.

A023416 := proc(n) local digs : digs := convert(n,base,2) : if nops(digs) = 0 then 1: else add(1-j,j=digs) : fi : end: ili := readline("b102370.txt") : while ili <> 0 do na := sscanf(ili,"%d %d") : na := A023416(op(2,na)) ; printf("%d, ",na) ; ili := readline("b102370.txt") : od: # R. J. Mathar, Aug 10 2007

a(n) = A023416(A102370(n)). - R. J. Mathar, Aug 10 2007

More terms from R. J. Mathar, Aug 10 2007

A105104 Write A102370 in binary (A103542), read backwards, omit leading zeros, convert to base 10.

Original entry on oeis.org

0, 3, 3, 5, 1, 15, 5, 9, 1, 13, 7, 11, 7, 29, 9, 17, 1, 25, 13, 21, 5, 31, 11, 19, 3, 27, 15, 47, 13, 57, 17, 33, 1, 49, 25, 41, 9, 61, 21, 37, 5, 53, 29, 45, 15, 59, 19, 35, 3, 51, 27, 43, 11, 63, 23, 39, 7, 55, 63, 93, 25, 113, 33, 65, 1, 97, 49, 81, 17, 121, 41, 73

Offset: 0

N. J. A. Sloane, Apr 30 2005

Similar to A102370, but read diagonals in reverse direction.

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

Cf. A030101, A102370, A103542.

a1(n) = fromdigits(Vecrev(binary(n)), 2); \\ A030101
a0(n) = if( n<1, 0, sum(k=0, length(binary(n)), bitand(n + k, 2^k))); \\ A102370
a(n) = a1(a0(n)); \\ Michel Marcus, Apr 09 2022

a(n) = A030101(A102370(n)). - Philippe Deléham, Nov 11 2007

More terms from Philippe Deléham, Nov 11 2007

a(46)=19 inserted and more terms from Georg Fischer, Apr 08 2022

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

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

A103615 Number of zeros in A103542(n) (binary equivalent of A102370(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A105104 Write A102370 in binary (A103542), read backwards, omit leading zeros, convert to base 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions