A103185 -id:A103185 - cp's OEIS frontend

A105031 Binary equivalents of A103185.

0, 1, 10, 1, 0, 101, 10, 1, 0, 1, 10, 1, 1000, 101, 10, 1, 0, 1, 10, 1, 0, 101, 10, 1, 0, 1, 10, 10001, 1000, 101, 10, 1, 0, 1, 10, 1, 0, 101, 10, 1, 0, 1, 10, 1, 1000, 101, 10, 1, 0, 1, 10, 1, 0, 101, 10, 1, 0, 1, 100010, 10001, 1000, 101, 10, 1, 0, 1, 10, 1, 0, 101, 10, 1, 0, 1, 10

Offset: 0

N. J. A. Sloane, David Applegate, Benoit Cloitre and Philippe Deléham, Apr 03 2005

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. A102370, A104234, A105032.

More terms from Benoit Cloitre, Apr 04 2005

A105023 a(n) = A102370(n) - n. Or, 2*A103185(n).

Original entry on oeis.org

0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4, 2, 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4, 34, 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4, 2, 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 68, 34, 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4, 2, 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4, 34, 16, 10, 4

Offset: 0

N. J. A. Sloane, Apr 03 2005

When written in base 2 as a right justified table, columns have periods 1, 2, 4, 8, ... - Philippe Deléham, Apr 21 2005

			Has a natural decomposition into blocks: 0; 2; 4, 2, 0; 10, 4, 2, 0, 2, 4, 2; 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4; 34, 16, 10, 4, ... where the leading term in each block is given by A105024.

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. A102370, A103185, A105024.

s:= proc (n) local t1, l; t1 := 0; for l to n do if `mod`(n+l,2^l) = 0 then t1 := t1+2^l end if end do; t1 end proc;

a(n) = Sum_{ k >= 1 such that n + k == 0 mod 2^k } 2^k.

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

A104234 Number of k >= 1 such that k+n == 0 mod 2^k.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 2, 2, 1, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1

Offset: 0

N. J. A. Sloane, Apr 02 2005

nonn

Number of terms in the summation in the formula for A102370(n).

Also, a(n) is the number of 1's in (A103185(n) written in base 2).

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. A102370, A103185, A105035 (records).

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

a(2^k + y) = a(y) + 1 if y = 2^k - k - 1, = a(y) otherwise (where 0 <= y <= 2^k - 1).

A103745 a(n) = (A102371(n) + n)/2.

Original entry on oeis.org

1, 2, 5, 8, 17, 34, 65, 128, 257, 514, 1029, 2048, 4097, 8194, 16385, 32768, 65537, 131074, 262149, 524296, 1048577, 2097154, 4194305, 8388608, 16777217, 33554434, 67108869, 134217728, 268435457, 536870914, 1073741825, 2147483648, 4294967297, 8589934594, 17179869189

Offset: 1

Philippe Deléham, Mar 26 2005

nonn

Values of A103185 (first zero omitted) which are >= a new power of 2 . The initial values of A103185 are 0*, 1*, 2*, 1, 0, 5*, 2, 1, 0, 1, 2, 1, 8*, 5, 2, 1, ... and the starred terms are those which exceed the next power of 2 . Their indices (except for the zero term) are given by A000325.

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. A000325, A102371, A103185, A103528.

a(n) = 2^(n-1) + sum(k = 1, n-1, if ((n % 2^k) == k, 2^(k-1))); \\ Michel Marcus, May 06 2020

a(n) = Sum_{ k>= 1, k == n (mod 2^k) } 2^(k-1). - N. J. A. Sloane and David Applegate, Mar 22 2005

a(n) = A103528(n) + 2^(n-1).

a(27) corrected and more terms from Michel Marcus, May 06 2020

cp's OEIS Frontend

A105031 Binary equivalents of A103185.

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A105023 a(n) = A102370(n) - n. Or, 2*A103185(n).

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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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A104234 Number of k >= 1 such that k+n == 0 mod 2^k.

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A103745 a(n) = (A102371(n) + n)/2.

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