A243071 -id:A243071 - cp's OEIS frontend

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

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).

A252735 a(1) = 0; for n > 1: a(2n) = a(n), a(2n+1) = 1 + a(A064989(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 0, 1, 2, 4, 1, 5, 3, 2, 0, 6, 1, 7, 2, 3, 4, 8, 1, 2, 5, 1, 3, 9, 2, 10, 0, 4, 6, 3, 1, 11, 7, 5, 2, 12, 3, 13, 4, 2, 8, 14, 1, 3, 2, 6, 5, 15, 1, 4, 3, 7, 9, 16, 2, 17, 10, 3, 0, 5, 4, 18, 6, 8, 3, 19, 1, 20, 11, 2, 7, 4, 5, 21, 2, 1, 12, 22, 3, 6, 13, 9, 4, 23, 2, 5, 8, 10, 14, 7, 1, 24, 3, 4, 2, 25, 6, 26, 5, 3, 15, 27, 1

Offset: 1

Antti Karttunen, Dec 21 2014

nonn

Consider the binary tree illustrated in A005940: If we start from any n, computing successive iterations of A252463 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located at), a(n) gives the number of odd numbers > 1 encountered on the path (i.e., excluding the final 1 from the count but including the starting n if it was odd).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Essentially one less than A061395.
Cf. A252464, A252736.
Cf. A000120, A005940, A064989, A080791, A156552, A243071, A252462, A252464, A252736.
Cf. also A246369.

a252735[n_] := Prepend[Rest@Array[PrimePi[FactorInteger[#][[-1]][[1]]] - 1 &, n], 0]; a252735[108] (* Michael De Vlieger, Dec 21 2014, after Stefan Steinerberger at A061395 *)

a(1) = 0; for n > 1: a(2n) = a(n), a(2n+1) =  1 + a(A064989(n)).
a(n) = A080791(A156552(n)). [Number of nonleading 0-bits in A156552(n).]
Other identities:
For all n >= 2:
a(n) = A061395(n) - 1.
a(n) = A000120(A243071(n)) - 1. [One less than the binary weight of A243071(n).]
a(n) = A252464(n) - A252736(n) - 1.

A364496 Numbers k such that k is a multiple of A163511(k).

Original entry on oeis.org

0, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 16383, 24576, 32766, 49152, 65532, 98304, 131064, 196608, 262128, 393216, 524256, 786432, 1048512, 1572864, 2097024, 3145728, 4194048, 6291456, 8388096, 12582912, 16776192, 25165824, 33552384, 50331648, 67104768, 100663296, 134209536, 201326592

Offset: 1

Antti Karttunen, Jul 27 2023

nonn

If n is present, then 2*n is also present, and vice versa.
A007283 is included as a subsequence, because it gives the known fixed points of map n -> A163511(n).
Sequence A243071(A364497(.)) sorted into ascending order.

			16383 is present, because A163511(16383) = 43, as 16383 = 2^14 - 1 and A000040(14) = 43, and 43 is a factor of 16383 = 3*43*127.
536870895 is present, because A163511(536870895) = 1177 (11*107), which divides 536870895 (3*5*11*47*107*647). See also example in A364498.

Antti Karttunen, Table of n, a(n) for n = 1..53

Positions of 1's in A364492.
Subsequence of A364292.
Cf. A007283 (subsequence), A163511, A364963 (odd terms).
Cf. also A364295, A364494, A364497, A364498.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1);
A163511(n) = if(!n,1,A005940(1+A054429(n)))
isA364496(n) = !(n%A163511(n));

A246376 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(n), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 13, 12, 21, 18, 11, 16, 25, 14, 33, 20, 15, 26, 29, 24, 17, 42, 19, 36, 53, 22, 73, 32, 43, 50, 37, 28, 45, 66, 31, 40, 57, 30, 81, 52, 27, 58, 61, 48, 49, 34, 35, 84, 117, 38, 41, 72, 87, 106, 169, 44, 213, 146, 67, 64, 65, 86, 89, 100, 91, 74, 173, 56, 149, 90, 51, 132, 101, 62, 113, 80, 23

Offset: 1

Antti Karttunen, Aug 27 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A246375.
Similar or related permutations: A005940, A005941, A064216, A243071, A245605, A246377, A246380.
Cf. A000035, A064989.

a(1) = 1, a(2n) = 2 * a(n), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).
As a composition of related permutations:
a(n) = A246377(A246380(n)).
Other identities. For all n >= 1 the following holds:
A000035(a(n)) = A000035(n). [Like A005940 & A005941, this also preserves the parity].

A252736 a(1) = a(2) = 0; for n > 2: a(2n) = 1 + a(n), a(2n+1) = a(A064989(2n+1)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 21 2014

nonn

Consider the binary tree illustrated in A005940: If we start from any n, computing successive iterations of A252463 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located), a(n) gives the number of even numbers > 2 encountered on the path (i.e., excluding the 2 from the count but including the starting n if it was even).

The number of pairs in any factorization tree of n. For example, a possible factorization tree of 12 is 12 -> (4*3) -> (2*2)*3. There are 2 pairs in this factor tree: (4*3) and (2*2). Thus, a(12) - 1 = 3 - 1 = 2. - Melvin Peralta, Aug 29 2016

Antti Karttunen, Table of n, a(n) for n = 1..8192

Essentially one less than A001222.
Cf. A000120, A005940, A064989, A080791, A156552, A243071, A252464, A252735.
Cf. also A246370.

a[1] = a[2] = 0; a[n_] := a[n] = If[EvenQ@ n, 1 + a[n/2], a[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n]]; Array[a, 120] (* Michael De Vlieger, Aug 30 2016 *)

a(1) = a(2) = 0; for n > 2: a(2n) = 1 + a(n), a(2n+1) =  a(A064989(2n+1)).
a(n) = A080791(A243071(n)). [Number of nonleading 0-bits in A243071(n).]
Other identities. For all n >= 2:
a(n) = A000120(A156552(n)) - 1. [One less than the binary weight of A156552(n).]
a(n) = A252464(n) - A252735(n) - 1.
a(n) = A001222(n) - 1.

A279339 a(1) = 1; for n > 1, if n is even, a(n) = A055938(a(n/2)), otherwise a(n) = A005187(a(A064989(n))).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 12, 8, 9, 11, 13, 19, 14, 10, 27, 35, 17, 67, 20, 16, 24, 131, 28, 15, 40, 22, 29, 259, 21, 515, 58, 25, 72, 18, 36, 1027, 136, 46, 43, 2051, 33, 4099, 51, 23, 264, 8195, 59, 26, 30, 78, 83, 16387, 45, 31, 60, 142, 520, 32771, 44, 65539, 1032, 38, 121, 47, 52, 131075, 147, 270, 37, 262147, 75, 524291, 2056, 32, 275, 34, 93

Offset: 1

Antti Karttunen, Dec 10 2016

nonn

A more recursed variant of A279337.

Antti Karttunen, Table of n, a(n) for n = 1..141
Index entries for sequences that are permutations of the natural numbers

Inverse: A279338.
Cf. A005187, A055938, A064989.
Related or similar permutations: A156552, A243071, A279337, A279342, A279344.

(definec (A279339 n) (cond ((= 1 n) n) ((even? n) (A055938 (A279339 (/ n 2)))) (else (A005187 (A279339 (A064989 n))))))

a(1) = 1; for n > 1, if n is even, a(n) = A055938(a(n/2)), otherwise a(n) = A005187(a(A064989(n))).
As a composition of other permutations:
a(n) = A279342(A243071(n)).
a(n) = A279344(A156552(n)).

A292253 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2a(A252463(n)) + [n == 1 (mod 2)][J(3|n) == 1], where J is the Jacobi-symbol.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 8, 4, 4, 8, 17, 8, 35, 16, 8, 8, 70, 8, 140, 16, 16, 34, 281, 16, 9, 70, 8, 32, 562, 16, 1124, 16, 32, 140, 17, 16, 2249, 280, 68, 32, 4498, 32, 8996, 68, 16, 562, 17993, 32, 19, 18, 140, 140, 35986, 16, 32, 64, 280, 1124, 71973, 32, 143947, 2248, 32, 32, 64, 64, 287894, 280, 560, 34, 575789, 32, 1151579, 4498, 16, 560, 34, 136, 2303158, 64

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

Base-2 expansion of a(n) encodes the steps where numbers that are either of the form 12k+1 or of the form 12k+11 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n. An exception is the most significant bit of a(n) which corresponds with the final 1, but is shifted one bit-position towards right.

The AND - XOR formula(s) just restate the fact that J(3|n) = J(-1|n)*J(-3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A243071, A292254, A292255, A292263, A292941, A292943, A292383.

(define (A292253 n) (if (<= n 2) (- n 1) (+ (if (and (odd? n) (= 1 (jacobi-symbol 3 n))) 1 0) (* 2 (A292253 (A252463 n))))))

a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == 1], where J is the Jacobi-symbol, and [ ]'s are Iverson brackets, whose product gives 1 only if n is an odd number for which J(3|n) = +1, and 0 otherwise.
a(n) = A292263(n) AND (A292383(n) XOR A292941(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987). [See comments.]
For n >= 0, a(A163511(n)) = A292254(n).
For n >= 1, a(n) + A292255(n) + A292943(n) = A243071(n).

A292254 a(n) = A292253(A163511(n)).

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 9, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 18, 19, 16, 16, 16, 17, 16, 16, 16, 17, 32, 32, 32, 33, 32, 32, 32, 32, 32, 32, 32, 32, 36, 36, 38, 39, 32, 32, 32, 32, 32, 32, 34, 34, 32, 32, 32, 32, 32, 32, 34, 35, 64, 64, 64, 64, 64, 64, 66, 67, 64, 64, 64, 65, 64, 64, 64, 65, 64, 64, 64, 65, 64, 64, 64, 64, 72, 72, 72, 72, 76, 76

Offset: 0

Antti Karttunen, Sep 28 2017

nonn

Because A292253(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 the numbers that are either of the form 12k+1 or of the form 12k+11 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).

The AND - XOR formula just restates the fact that J(3|n) = J(-1|n)*J(-3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A292253.

Cf also A292247, A292248, A292256, A292264, A292271, A292274, A292592, A292593, A292942, A292944, A292946 (for similarly constructed sequences).

(define (A292254 n) (A292253 (A163511 n)))

a(n) = A292253(A163511(n)).
a(n) = A292264(n) AND (A292274(n) XOR A292942(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987). [See comments.]
For all n >= 0, a(n) + A292944(n) + A292256(n) = n.

A292255 a(1) = 0, and for n > 1, a(n) = 2a(A252463(n)) + [n == 1 (mod 2)][J(3|n) == -1], where J is the Jacobi-symbol.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 0, 0, 2, 6, 0, 12, 6, 0, 0, 25, 0, 51, 4, 4, 12, 102, 0, 0, 24, 0, 12, 205, 0, 411, 0, 12, 50, 0, 0, 822, 102, 24, 8, 1645, 8, 3291, 24, 0, 204, 6582, 0, 0, 0, 48, 48, 13165, 0, 9, 24, 100, 410, 26330, 0, 52660, 822, 8, 0, 25, 24, 105321, 100, 204, 0, 210642, 0, 421284, 1644, 0, 204, 1, 48, 842569, 16, 0, 3290, 1685138, 16, 48, 6582

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

Base-2 expansion of a(n) encodes the steps where numbers that are either of the form 12k+5 or of the form 12k+7 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

The AND - XOR formulas just restate the fact that J(3|n) = J(-1|n)*J(-3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A243071, A292253, A292256, A292263, A292943.

Cf. also A292941, A292945.

(define (A292255 n) (if (<= n 1) 0 (+ (if (and (odd? n) (= -1 (jacobi-symbol 3 n))) 1 0) (* 2 (A292255 (A252463 n))))))

a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == -1], where J is the Jacobi-symbol, and [ ]'s are Iverson brackets, whose product gives 1 only if n is an odd number for which J(3|n) = -1, and 0 otherwise.
a(n) = A292263(n) AND (A292383(n) XOR A292945(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
a(n) = A292263(n) AND (A292385(n) XOR A292941(n)). [See comments.]
For n >= 0, a(A163511(n)) = A292256(n).
For n >= 1, a(n) + A292253(n) + A292943(n) = A243071(n).

A292256 a(n) = A292255(A163511(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 6, 6, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 8, 8, 8, 9, 12, 12, 12, 12, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 6, 6, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 16, 16, 16, 16, 16, 16

Offset: 0

Antti Karttunen, Sep 28 2017

nonn

Because A292255(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 the numbers that are either of the form 12k+5 or of the form 12k+7 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).

The AND - XOR formulas just restate the fact that J(3|n) = J(-1|n)*J(-3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A292255.

Cf. also A292247, A292248, A292254, A292264, A292271, A292274, A292592, A292593, A292942, A292944, A292946 (for similarly constructed sequences).

(define (A292256 n) (A292255 (A163511 n)))

a(n) = A292255(A163511(n)).
a(n) = A292264(n) AND (A292274(n) XOR A292946(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
a(n) = A292264(n) AND (A292271(n) XOR A292942(n)). [See comments].
For all n >= 0, a(n) + A292944(n) + A292254(n) = n.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

A252735 a(1) = 0; for n > 1: a(2n) = a(n), a(2n+1) = 1 + a(A064989(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A364496 Numbers k such that k is a multiple of A163511(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A246376 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(n), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

Views

Author

Keywords

Links

Crossrefs

Formula

A252736 a(1) = a(2) = 0; for n > 2: a(2n) = 1 + a(n), a(2n+1) = a(A064989(2n+1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A279339 a(1) = 1; for n > 1, if n is even, a(n) = A055938(a(n/2)), otherwise a(n) = A005187(a(A064989(n))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A292253 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == 1], where J is the Jacobi-symbol.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A292254 a(n) = A292253(A163511(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A292255 a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(3|n) == -1], where J is the Jacobi-symbol.

A292253 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2a(A252463(n)) + [n == 1 (mod 2)][J(3|n) == 1], where J is the Jacobi-symbol.

A292255 a(1) = 0, and for n > 1, a(n) = 2a(A252463(n)) + [n == 1 (mod 2)][J(3|n) == -1], where J is the Jacobi-symbol.