A048735 -id:A048735 - cp's OEIS frontend

A292272 a(n) = n - A048735(n) = n - (n AND floor(n/2)).

0, 1, 2, 2, 4, 5, 4, 4, 8, 9, 10, 10, 8, 9, 8, 8, 16, 17, 18, 18, 20, 21, 20, 20, 16, 17, 18, 18, 16, 17, 16, 16, 32, 33, 34, 34, 36, 37, 36, 36, 40, 41, 42, 42, 40, 41, 40, 40, 32, 33, 34, 34, 36, 37, 36, 36, 32, 33, 34, 34, 32, 33, 32, 32, 64, 65, 66, 66, 68, 69, 68, 68, 72, 73, 74, 74, 72, 73, 72, 72, 80, 81, 82, 82, 84, 85, 84, 84, 80, 81, 82, 82, 80, 81

Offset: 0

Antti Karttunen, Sep 16 2017

In binary expansion of n, change those 1's to 0's that have an 1-bit next to them at their left (more significant) side. Only fibbinary numbers (A003714) occur as terms.

			From _Kevin Ryde_, Jun 02 2020: (Start)
     n = 1831 = binary 11100100111
  a(n) = 1060 = binary 10000100100   high 1 of each run
(End)

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A003188, A003714, A003987, A004198, A005940, A048735, A085357, A292371, A292382.

Table[n - BitAnd[n, Floor[n/2]], {n, 0, 93}] (* Michael De Vlieger, Sep 17 2017 *)

a(n) = bitnegimply(n,n>>1); \\ Kevin Ryde, Jun 02 2020

def A292272(n): return n&~(n>>1) # Chai Wah Wu, Jun 29 2022

a(n) = n - A048735(n) = n - (n AND floor(n/2)) = n XOR (n AND floor(n/2)), where AND is bitwise-AND (A004198) and XOR is bitwise-XOR (A003987).
a(n) = n AND A003188(n).
a(n) = A292382(A005940(1+n)).
A059905(a(n)) = A292371(n).
For all n >= 0, A085357(a(n)) = 1.
a(n) = A213064(n) / 2. - Kevin Ryde, Jun 02 2020
a(n) = n AND NOT floor(n/2). - Chai Wah Wu, Jun 29 2022

A156552 Unary-encoded compressed factorization of natural numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 128, 19, 18, 33, 256, 23, 12, 65, 14, 35, 512, 21, 1024, 31, 34, 129, 20, 27, 2048, 257, 66, 39, 4096, 37, 8192, 67, 22, 513, 16384, 47, 24, 25, 130, 131, 32768, 29, 36, 71, 258, 1025, 65536, 43, 131072, 2049, 38, 63, 68, 69, 262144

Offset: 1

Leonid Broukhis, Feb 09 2009

The primes become the powers of 2 (2 -> 1, 3 -> 2, 5 -> 4, 7 -> 8); the composite numbers are formed by taking the values for the factors in the increasing order, multiplying them by the consecutive powers of 2, and summing. See the Example section.
From Antti Karttunen, Jun 27 2014: (Start)
The odd bisection (containing even terms) halved gives A244153.
The even bisection (containing odd terms), when one is subtracted from each and halved, gives this sequence back.
(End)
Question: Are there any other solutions that would satisfy the recurrence r(1) = 0; and for n > 1, r(n) = Sum_{d|n, d>1} 2^A033265(r(d)), apart from simple variants 2^k * A156552(n)? See also A297112, A297113. - Antti Karttunen, Dec 30 2017

			For 84 = 2*2*3*7 -> 1*1 + 1*2 + 2*4 + 8*8 =  75.
For 105 = 3*5*7 -> 2*1 + 4*2 + 8*4 = 42.
For 137 = p_33 -> 2^32 = 4294967296.
For 420 = 2*2*3*5*7 -> 1*1 + 1*2 + 2*4 + 4*8 + 8*16 = 171.
For 147 = 3*7*7 = p_2 * p_4 * p_4 -> 2*1 + 8*2 + 8*4 = 50.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1024 terms from Antti Karttunen)
Hans Havermann, Factorization of the first 10000 terms, in format [[primes], [exponents]]
A puzzle by Sergey Orlov (in Russian)
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences computed from indices in prime factorization

One less than A005941.
Inverse permutation: A005940 with starting offset 0 instead of 1.
Cf. A000079, A000120, A001222, A052126, A054429, A061395, A064216, A064989, A003188, A243071, A243065-A243066, A244153, A243354, A112798, A125106, A056239, A161511.
Cf. also A297106, A297112 (Möbius transform), A297113, A153013, A290308, A300827, A323243, A323244, A323247, A324201, A324812 (n for which a(n) is a square), A324813, A324822, A324823, A324398, A324713, A324815, A324819, A324865, A324866, A324867.
Other related permutations: A253551, A253792, A253564, A253791, A277195, A297163, A297164, A297165, A297166, A302023, A305418, A322863, A322864.

Table[Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[ Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ n]], {n, 67}] (* Michael De Vlieger, Sep 08 2016 *)

a(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ David A. Corneth, Mar 08 2019

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n)))); \\ (based on the given recurrence) - Antti Karttunen, Mar 08 2019

# Program corrected per instructions from Leonid Broukhis. - Antti Karttunen, Jun 26 2014
# However, it gives correct answers only up to n=136, before corruption by a wrap-around effect.
# Note that the correct answer for n=137 is A156552(137) = 4294967296.
$max = $ARGV[0];
$pow = 0;
foreach $i (2..$max) {
@a = split(/ /, `factor $i`);
shift @a;
$shift = 0;
$cur = 0;
while ($n = int shift @a) {
$prime{$n} = 1 << $pow++ if !defined($prime{$n});
$cur |= $prime{$n} << $shift++;
}
print "$cur, ";
}
print "\n";
(Scheme, with memoization-macro definec from Antti Karttunen's IntSeq-library, two different implementations)
(definec (A156552 n) (cond ((= n 1) 0) (else (+ (A000079 (+ -2 (A001222 n) (A061395 n))) (A156552 (A052126 n))))))
(definec (A156552 n) (cond ((= 1 n) (- n 1)) ((even? n) (+ 1 (* 2 (A156552 (/ n 2))))) (else (* 2 (A156552 (A064989 n))))))
;; Antti Karttunen, Jun 26 2014

from sympy import primepi, factorint
def A156552(n): return sum((1<Chai Wah Wu, Mar 10 2023

From Antti Karttunen, Jun 26 2014: (Start)
a(1) = 0, a(n) = A000079(A001222(n)+A061395(n)-2) + a(A052126(n)).
a(1) = 0, a(2n) = 1+2*a(n), a(2n+1) = 2*a(A064989(2n+1)). [Compare to the entanglement recurrence A243071].
For n >= 0, a(2n+1) = 2*A244153(n+1). [Follows from the latter clause of the above formula.]
a(n) = A005941(n) - 1.
As a composition of related permutations:
a(n) = A003188(A243354(n)).
a(n) = A054429(A243071(n)).
For all n >= 1, A005940(1+a(n)) = n and for all n >= 0, a(A005940(n+1)) = n. [The offset-0 version of A005940 works as an inverse for this permutation.]
This permutations also maps between the partition-lists A112798 and A125106:
A056239(n) = A161511(a(n)). [The sums of parts of each partition (the total sizes).]
A003963(n) = A243499(a(n)). [And also the products of those parts.]
(End)
From Antti Karttunen, Oct 09 2016: (Start)
A161511(a(n)) = A056239(n).
A029837(1+a(n)) = A252464(n). [Binary width of terms.]
A080791(a(n)) = A252735(n). [Number of nonleading 0-bits.]
A000120(a(n)) = A001222(n). [Binary weight.]
For all n >= 2, A001511(a(n)) = A055396(n).
For all n >= 2, A000120(a(n))-1 = A252736(n). [Binary weight minus one.]
A252750(a(n)) = A252748(n).
a(A250246(n)) = A252754(n).
a(A005117(n)) = A277010(n). [Maps squarefree numbers to a permutation of A003714, fibbinary numbers.]
A085357(a(n)) = A008966(n). [Ditto for their characteristic functions.]
For all n >= 0:
a(A276076(n)) = A277012(n).
a(A276086(n)) = A277022(n).
a(A260443(n)) = A277020(n).
(End)
From Antti Karttunen, Dec 30 2017: (Start)
For n > 1, a(n) = Sum_{d|n, d>1} 2^A033265(a(d)). [See comments.]
More linking formulas:
A106737(a(n)) = A000005(n).
A290077(a(n)) = A000010(n).
A069010(a(n)) = A001221(n).
A136277(a(n)) = A181591(n).
A132971(a(n)) = A008683(n).
A106400(a(n)) = A008836(n).
A268411(a(n)) = A092248(n).
A037011(a(n)) = A010052(n) [conjectured, depends on the exact definition of A037011].
A278161(a(n)) = A046951(n).
A001316(a(n)) = A061142(n).
A277561(a(n)) = A034444(n).
A286575(a(n)) = A037445(n).
A246029(a(n)) = A181819(n).
A278159(a(n)) = A124859(n).
A246660(a(n)) = A112624(n).
A246596(a(n)) = A069739(n).
A295896(a(n)) = A053866(n).
A295875(a(n)) = A295297(n).
A284569(a(n)) = A072411(n).
A286574(a(n)) = A064547(n).
A048735(a(n)) = A292380(n).
A292272(a(n)) = A292382(n).
A244154(a(n)) = A048673(n), a(A064216(n)) = A244153(n).
A279344(a(n)) = A279339(n), a(A279338(n)) = A279343(n).
a(A277324(n)) = A277189(n).
A037800(a(n)) = A297155(n).
For n > 1, A033265(a(n)) = 1+A297113(n).
(End)
From Antti Karttunen, Mar 08 2019: (Start)
a(n) = A048675(n) + A323905(n).
a(A324201(n)) = A000396(n), provided there are no odd perfect numbers.
The following sequences are derived from or related to the base-2 expansion of a(n):
A000265(a(n)) = A322993(n).
A002487(a(n)) = A323902(n).
A005187(a(n)) = A323247(n).
A324288(a(n)) = A324116(n).
A323505(a(n)) = A323508(n).
A079559(a(n)) = A323512(n).
A085405(a(n)) = A323239(n).
The following sequences are obtained by applying to a(n) a function that depends on the prime factorization of its argument, which goes "against the grain" because a(n) is the binary code of the factorization of n, which in these cases is then factored again:
A000203(a(n)) = A323243(n).
A033879(a(n)) = A323244(n) = 2*a(n) - A323243(n),
A294898(a(n)) = A323248(n).
A000005(a(n)) = A324105(n).
A000010(a(n)) = A324104(n).
A083254(a(n)) = A324103(n).
A001227(a(n)) = A324117(n).
A000593(a(n)) = A324118(n).
A001221(a(n)) = A324119(n).
A009194(a(n)) = A324396(n).
A318458(a(n)) = A324398(n).
A192895(a(n)) = A324100(n).
A106315(a(n)) = A324051(n).
A010052(a(n)) = A324822(n).
A053866(a(n)) = A324823(n).
A001065(a(n)) = A324865(n) = A323243(n) - a(n),
A318456(a(n)) = A324866(n) = A324865(n) OR a(n),
A318457(a(n)) = A324867(n) = A324865(n) XOR a(n),
A318458(a(n)) = A324398(n) = A324865(n) AND a(n),
A318466(a(n)) = A324819(n) = A323243(n) OR 2*a(n),
A318467(a(n)) = A324713(n) = A323243(n) XOR 2*a(n),
A318468(a(n)) = A324815(n) = A323243(n) AND 2*a(n).
(End)

More terms from Antti Karttunen, Jun 28 2014

A213370 a(n) = n AND 2*n, where AND is the bitwise AND operator.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 4, 6, 0, 0, 0, 2, 8, 8, 12, 14, 0, 0, 0, 2, 0, 0, 4, 6, 16, 16, 16, 18, 24, 24, 28, 30, 0, 0, 0, 2, 0, 0, 4, 6, 0, 0, 0, 2, 8, 8, 12, 14, 32, 32, 32, 34, 32, 32, 36, 38, 48, 48, 48, 50, 56, 56, 60, 62, 0, 0, 0, 2, 0, 0, 4, 6, 0, 0, 0, 2, 8, 8

Offset: 0

Alex Ratushnyak, Jun 14 2012

Antti Karttunen, Table of n, a(n) for n = 0..16384 (terms 0..1023 from T. D. Noe)
Index entries for sequences related to binary expansion of n

Cf. A080099, A213526, A048727, A048735, A269160.
Cf. A003714: indices of 0's.
Cf. A213540: indices of 2's, indices of 4's divided by 2.

Table[BitAnd[n, 2n], {n, 0, 63}] (* Alonso del Arte, Jun 19 2012 *)

a(n) = bitand(n, 2*n); \\ Michel Marcus, Mar 26 2021

for n in range(99):
    print(2*n & n, end=", ")

a(n) = 2 * A048735(n).

a(n) = (1/2)*(A048727(n) XOR A269160(n)) = (n OR 2n) XOR (n XOR 2n). - Antti Karttunen, May 16 2021

A292944 a(n) = A292272(A004754(n)) - 2*A053644(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 2, 0, 1, 2, 2, 4, 5, 4, 4, 0, 1, 2, 2, 4, 5, 4, 4, 8, 9, 10, 10, 8, 9, 8, 8, 0, 1, 2, 2, 4, 5, 4, 4, 8, 9, 10, 10, 8, 9, 8, 8, 16, 17, 18, 18, 20, 21, 20, 20, 16, 17, 18, 18, 16, 17, 16, 16, 0, 1, 2, 2, 4, 5, 4, 4, 8, 9, 10, 10, 8, 9, 8, 8, 16, 17, 18, 18, 20, 21, 20, 20, 16, 17, 18, 18, 16, 17, 16, 16, 32, 33, 34, 34, 36, 37, 36, 36

Offset: 0

Antti Karttunen, Sep 28 2017

In binary expansion (A007088) of n, clear the most significant bit and all those 1-bits that have another 1-bit at their left side, except for the second most significant 1-bit, even in cases where the binary expansion begins as "11...".

Because A292943(n) = a(A243071(n)), the sequence works as a "masking function" where the 1-bits in a(n) (always a subset of the 1-bits in binary expansion of n) indicate which numbers are of the form 6k+3 (odd multiples of three) in binary tree A163511 (or its mirror image tree A005940) on that trajectory which leads from the root of the tree to the node containing A163511(n).

			For n = 23, 10111 in binary, when we clear (change to zero) the most significant bit (always 1) and also all 1-bits that have 1's at their left side, we are left with 100, which in binary stands for 4, thus a(23) = 4.
For n = 27, 11011 in binary, when we clear the most significant bit, and also all 1-bits that have 1's at their left side except the second most significant, we are left with 1010, which in binary stands for ten, thus a(27) = 10.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A004754, A005940, A048735, A163511, A292272, A292943.

Cf. also A292247, A292248, A292254, A292256, A292264, A292271, A292274, A292592, A292593, A292942, A292946.

(define (A292944 n) (let ((x (+ n (A053644 n)))) (- (A292272 x) (A053644 x))))
(define (A292944 n) (- (A292272 (A004754 n)) (* 2 (A053644 n))))
(define (A292944 n) (A292943 (A163511 n)))

a(n) = A292272(A004754(n)) - 2*A053644(n).
a(n) = A292943(A163511(n)).
Other identities. For all n >= 0:
a(n) + A292264(n) = A292942(n) + a(n) + A292946(n) = a(n) + A292254(n) + A292256(n) = n.
a(n) = a(n) AND n; a(n) AND A292264(n) = 0, where AND is bitwise-and (A004198).

A048728 Differences between A008585 (multiples of 3) and A048724.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 8, 12, 0, 0, 0, 4, 16, 16, 24, 28, 0, 0, 0, 4, 0, 0, 8, 12, 32, 32, 32, 36, 48, 48, 56, 60, 0, 0, 0, 4, 0, 0, 8, 12, 0, 0, 0, 4, 16, 16, 24, 28, 64, 64, 64, 68, 64, 64, 72, 76, 96, 96, 96, 100, 112, 112, 120

Offset: 0

Antti Karttunen, Apr 26 1999

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Positions of zeros are given by A003714. Cf. A048735, A242400.

Diagonal 3 of A061858.

a048728 n = a008585 n - a048724 n  -- Reinhard Zumkeller, May 13 2014

a(n)=3*n - bitxor(2*n, n) \\ Charles R Greathouse IV, Oct 03 2016

def a(n): return 3*n - (2*n^n) # Indranil Ghosh, Jun 13 2017

a(n) = n*3 - Xmult(n, 3).

A292264 a(n) = n - A292944(n).

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 5, 8, 8, 8, 9, 8, 8, 10, 11, 16, 16, 16, 17, 16, 16, 18, 19, 16, 16, 16, 17, 20, 20, 22, 23, 32, 32, 32, 33, 32, 32, 34, 35, 32, 32, 32, 33, 36, 36, 38, 39, 32, 32, 32, 33, 32, 32, 34, 35, 40, 40, 40, 41, 44, 44, 46, 47, 64, 64, 64, 65, 64, 64, 66, 67, 64, 64, 64, 65, 68, 68, 70, 71, 64, 64, 64, 65, 64, 64, 66, 67, 72, 72, 72, 73, 76, 76

Offset: 0

Antti Karttunen, Sep 30 2017

Because A292263(n) = a(A243071(n)), the sequence works as a "masking function" where the 1-bits in a(n) (always a subset of the 1-bits in binary expansion of n) indicate which numbers are of the form 6k+1 or 6k+5 in binary tree A163511 (or in its mirror image tree A005940) on that trajectory which leads from the root of the tree to the node containing A163511(n).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A005940, A163511, A292263.

Cf. A048735, A292944, A292272 but also A292254, A292256, A292942, A292946 for similarly constructed sequences.

(define (A292264 n) (A292263 (A163511 n)))

a(n) = n - A292944(n).
a(n) = A292263(A163511(n)).
a(n) = A292942(n) + A292946(n).
a(n) = A292254(n) + A292256(n).

A292380 Base-2 expansion of a(n) encodes the steps where multiples of 4 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 0, 1, 0, 0, 0, 7, 0, 4, 0, 1, 0, 0, 0, 3, 4, 0, 6, 1, 0, 0, 0, 15, 0, 0, 0, 9, 0, 0, 0, 3, 0, 0, 0, 1, 2, 0, 0, 7, 8, 8, 0, 1, 0, 12, 0, 3, 0, 0, 0, 1, 0, 0, 2, 31, 0, 0, 0, 1, 0, 0, 0, 19, 0, 0, 8, 1, 0, 0, 0, 7, 14, 0, 0, 1, 0, 0, 0, 3, 0, 4, 0, 1, 0, 0, 0, 15, 0, 16, 2, 17, 0, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Sep 15 2017

			For n = 4, the starting value is a multiple of four, after which follows A252463(4) = 2, and A252463(2) = 1, the end point of iteration, and neither 2 nor 1 is a multiple of four, thus a(4) = 1*(2^0) + 0*(2^1) + 0*(2^2) = 1.
For n = 8, the starting value is a multiple of four, after which follows A252463(8) = 4 (also a multiple), continuing as before as 4 -> 2 -> 1, thus a(8) = 1*(2^0) + 1*(2^1) + 0*(2^2) + 0*(2^3) = 3.
For n = 9, the starting value is not a multiple of four, after which follows A252463(9) = 4 (which is), continuing as before as 4 -> 2 -> 1, thus a(9) = 0*(2^0) + 1*(2^1) + 0*(2^2) + 0*(2^3) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A005940, A048735, A156552, A292370, A292381, A292382, A292383, A292384.

Table[FromDigits[Reverse@ NestWhileList[Function[k, Which[k == 1, 1, EvenQ@ k, k/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ k]], n, # > 1 &] /. k_ /; IntegerQ@ k :> If[Mod[k, 4] == 0, 1, 0], 2], {n, 105}] (* Michael De Vlieger, Sep 21 2017 *)

a(n) = my(m=factor(n),k=-1,ret=0); for(i=1,matsize(m)[1], ret += bitneg(0,m[i,2]-1) << (primepi(m[i,1])+k); k+=m[i,2]); ret; \\ Kevin Ryde, Dec 11 2020

from sympy.core.cache import cacheit
from sympy.ntheory.factor_ import digits
from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a292370(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join(['1' if i==0 else '0' for i in k]), 2)
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a252463(n): return 1 if n==1 else n//2 if n%2==0 else a064989(n)
@cacheit
def a292384(n): return 1 if n==1 else 4*a292384(a252463(n)) + n%4
def a(n): return a292370(a292384(n))
print([a(n) for n in range(1, 111)]) # Indranil Ghosh, Sep 21 2017

(define (A292380 n) (A292370 (A292384 n)))

a(n) = A048735(A156552(n)).
a(n) = A292370(A292384(n)).
Other identities. For n >= 1:
a(n) AND A292382(n) = 0, where AND is a bitwise-AND (A004198).
a(n) + A292382(n) = A156552(n).
A000120(a(n)) + A000120(A292382(n)) = A001222(n).
A000035(a(n)) = A121262(n).

A292373 A binary encoding of 3-digits in base-4 representation of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 6, 6, 6, 7, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 5, 4, 4, 4, 5, 4

Offset: 0

Antti Karttunen, Sep 15 2017

			   n      a(n)     base-4(n)  binary(a(n))
                  A007090(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        0          1           0
   2        0          2           0
   3        1          3           1
   4        0         10           0
   5        0         11           0
   6        0         12           0
   7        1         13           1
   8        0         20           0
   9        0         21           0
  10        0         22           0
  11        1         23           1
  12        2         30          10
  13        2         31          10
  14        2         32          10
  15        3         33          11
  16        0        100           0
  17        0        101           0
  18        0        102           0
  19        1        103           1

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A007088, A007090, A048735, A059905, A059906, A160383, A213370, A292370, A292371, A292372.

def A292373(n): return int(bin(n&n>>1)[:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

a(n) = A059905(A048735(n)) = A059906(A213370(n)).

For all n >= 0, A000120(a(n)) = A160383(n).

A126306 a(n) = number of double-rises (UU-subsequences) in the n-th Dyck path encoded by A014486(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 3, 0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 3, 1, 2, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 3, 4, 0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 3, 1, 2, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 3

Offset: 0

Antti Karttunen, Jan 02 2007

nonn

			A014486(20) = 228 (11100100 in binary), encodes the following Dyck path:
    /\
   /..\/\
  /......\
and there is one rising (left-hand side) slope with length 3 and one with length 1, so in the first slope, consisting of 3 U-steps, there are two cases with two consecutive U-steps (overlapping is allowed), thus a(20)=2.

Cf. A014081, A014486, A000120, A048735, A125976, A057514.

def ok(n):
    if n==0: return True
    B=bin(n)[2:] if n!=0 else '0'
    s=0
    for b in B:
        s+=1 if b=='1' else -1
        if s<0: return False
    return s==0
def a014081(n): return sum(((n>>i)&3==3) for i in range(len(bin(n)[2:]) - 1))
print([a014081(n) for n in range(4001) if ok(n)]) # Indranil Ghosh, Jun 13 2017

a(n) = A014081(A014486(n)).
a(n) = A000120(A048735(A014486(n))).
a(A125976(n)) = A057514(n)-1, for all n >= 1.

A229762 a(n) = (n XOR floor(n/2)) AND floor(n/2), where AND and XOR are bitwise logical operators.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 1, 0, 4, 4, 5, 4, 2, 2, 1, 0, 8, 8, 9, 8, 10, 10, 9, 8, 4, 4, 5, 4, 2, 2, 1, 0, 16, 16, 17, 16, 18, 18, 17, 16, 20, 20, 21, 20, 18, 18, 17, 16, 8, 8, 9, 8, 10, 10, 9, 8, 4, 4, 5, 4, 2, 2, 1, 0, 32, 32, 33, 32, 34, 34, 33, 32, 36, 36, 37, 36, 34, 34, 33

Offset: 0

Alex Ratushnyak, Sep 28 2013

a(n) has a 01 bit pair in place of each 10 bit pair in n, and everywhere else 0 bits. Or equivalently a(n) has a 1-bit immediately below each run of 1's in n, but excluding a run ending at the least significant bit since below that is below the radix point. - Kevin Ryde, Feb 27 2021

			From _Kevin Ryde_, Feb 27 2021: (Start)
     n = 7267 = binary 1110001100011
  a(n) =  528 = binary   01000010000   1-bit below each run
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191

Cf. A003188 (n XOR floor(n/2)).
Cf. A048724 (n XOR (n*2)).
Cf. A048735 (n AND floor(n/2)).
Cf. A213370 (n AND (n*2)).
Cf. A213064 (n XOR (n*2) AND (n*2), 1-bit above each run).
Cf. A229763 ((2*n) XOR n AND n, low 1-bit each run).

import Data.Bits ((.&.), xor, shiftR)
a229762 n = (n `xor` shiftR n 1) .&. shiftR n 1 :: Int
-- Reinhard Zumkeller, Oct 10 2013

a(n) = bitnegimply(n>>1,n); \\ Kevin Ryde, Feb 27 2021

for n in range(333): print (n ^ (n>>1)) & (n>>1),

def A229762(n): return ~n& n>>1 # Chai Wah Wu, Jun 29 2022

a(n) = (n XOR floor(n/2)) AND floor(n/2) = (n AND floor(n/2)) XOR floor(n/2).

a(n) = floor(n/2) AND NOT n. - Chai Wah Wu, Jun 29 2022

cp's OEIS Frontend

A292272 a(n) = n - A048735(n) = n - (n AND floor(n/2)).

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A156552 Unary-encoded compressed factorization of natural numbers.

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A213370 a(n) = n AND 2*n, where AND is the bitwise AND operator.

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A292944 a(n) = A292272(A004754(n)) - 2*A053644(n).

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A048728 Differences between A008585 (multiples of 3) and A048724.

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A292264 a(n) = n - A292944(n).

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A292380 Base-2 expansion of a(n) encodes the steps where multiples of 4 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

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