A005187 -id:A005187 - cp's OEIS frontend

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

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)).

A300244 Difference between A005187 and its Möbius transform (A297111).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 10, 1, 14, 1, 13, 11, 15, 1, 22, 1, 22, 14, 21, 1, 30, 8, 25, 16, 29, 1, 40, 1, 31, 22, 34, 18, 46, 1, 37, 26, 46, 1, 57, 1, 45, 38, 44, 1, 62, 11, 57, 35, 53, 1, 68, 26, 61, 38, 56, 1, 84, 1, 59, 51, 63, 30, 90, 1, 70, 45, 89, 1, 94, 1, 73, 65, 77, 29, 104, 1, 94, 50, 81, 1, 117, 39, 84, 57, 93, 1, 128, 33, 92, 60, 91, 42

Offset: 1

Antti Karttunen, Mar 10 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005187, A008683, A297111, A297114, A297117.

Table[IntegerExponent[(2 n)!, 2] - DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 95}] (* or *)
Fold[Function[{a, n}, Append[a, {Abs@ Total@ Map[MoebiusMu[n/#] a[[#, -1]] &, Most@ Divisors@ n], IntegerExponent[(2 n)!, 2]}]], {{0, 1}}, Range[2, 95]][[All, 1]] (* Michael De Vlieger, Mar 10 2018 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A300244(n) = -sumdiv(n,d,(dA005187(d));

a(n) = A005187(n) - A297111(n).

a(n) = -Sum_{d|n, dA008683(n/d)*A005187(d).

A279342 a(0) = 1, a(1) = 2, a(2n) = A055938(a(n)), a(2n+1) = A005187(a(n)).

Original entry on oeis.org

1, 2, 5, 3, 12, 8, 6, 4, 27, 22, 17, 15, 13, 10, 9, 7, 58, 50, 45, 41, 36, 32, 30, 26, 28, 23, 21, 18, 20, 16, 14, 11, 121, 112, 103, 97, 92, 86, 84, 79, 75, 70, 65, 63, 61, 56, 55, 49, 59, 53, 48, 42, 44, 39, 37, 34, 43, 38, 33, 31, 29, 25, 24, 19, 248, 237, 227, 221, 210, 201, 196, 191, 187, 180, 175, 168, 171, 165, 160, 153, 154, 146, 141

Offset: 0

Antti Karttunen, Dec 10 2016

Note the indexing: the domain starts from 0, while the range excludes zero.
This sequence can be represented as a binary tree. Each left hand child is produced as A055938(n), and each right hand child as A005187(n), when the parent node contains n:
                                     1
                                     |
                  ...................2...................
                 5                                       3
      12......../ \........8                   6......../ \........4
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  27       22         17       15         13       10          9       7
58  50   45  41     36  32   30  26     28  23   21  18      20 16   14 11
etc.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A279341.
Right edge: A256994.
Cf. A005187, A055938.
Related or similar permutations: A054429, A163511, A233278, A256997, A279339, A279344, A279347.

(definec (A279342 n) (cond ((<= n 1) (+ 1 n)) ((even? n) (A055938 (A279342 (/ n 2)))) (else (A005187 (A279342 (/ (- n 1) 2))))))

a(0) = 1, a(1) = 2, and then after, a(2n) = A055938(a(n)), a(2n+1) = A005187(a(n)).
As a composition of other permutations:
a(n) = A279344(A054429(n)).
a(n) = A279347(A279344(n)).
a(n) = A279339(A163511(n)).

A279344 a(0) = 1, a(2n) = A005187(a(n)), a(2n+1) = A055938(a(n)).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 12, 7, 9, 10, 13, 15, 17, 22, 27, 11, 14, 16, 20, 18, 21, 23, 28, 26, 30, 32, 36, 41, 45, 50, 58, 19, 24, 25, 29, 31, 33, 38, 43, 34, 37, 39, 44, 42, 48, 53, 59, 49, 55, 56, 61, 63, 65, 70, 75, 79, 84, 86, 92, 97, 103, 112, 121, 35, 40, 46, 51, 47, 52, 54, 60, 57, 62, 64, 68, 73, 77, 82, 90, 66, 69, 71, 76, 74, 80

Offset: 0

Antti Karttunen, Dec 10 2016

Note the indexing: the domain starts from 0, while the range excludes zero.
This sequence can be represented as a binary tree. Each left hand child is produced as A005187(n), and each right hand child as A055938(n), when the parent node contains n:
                                    1
                                    |
                 ...................2...................
                3                                       5
      4......../ \........6                   8......../ \........12
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  7       9          10       13         15       17         22       27
11 14   16 20      18  21   23  28     26  30   32  36     41  45   50  58
etc.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A279343.
Left edge: A256994.
Cf. A005187, A055938.
Related or similar permutations: A005940, A054429, A233276, A256997, A279339, A279342, A279347.

(definec (A279344 n) (cond ((zero? n) 1) ((even? n) (A005187 (A279344 (/ n 2)))) (else (A055938 (A279344 (/ (- n 1) 2))))))

a(0) = 1, after which, a(2n) = A005187(a(n)), a(2n+1) = A055938(a(n)).
As a composition of other permutations:
a(n) = A279342(A054429(n)).
a(n) = A279347(A279342(n)).
a(n) = A279339(A005940(1+n)).

A279347 Self-inverse permutation of natural numbers: a(1) = 1, a(2) = 2, a(A055938(n)) = A005187(a(n)), a(A005187(n)) = A055938(a(n)).

Original entry on oeis.org

1, 2, 5, 12, 3, 8, 27, 6, 22, 17, 58, 4, 15, 50, 13, 45, 10, 36, 121, 41, 32, 9, 30, 112, 103, 28, 7, 26, 97, 23, 92, 21, 86, 75, 248, 18, 70, 84, 65, 237, 20, 61, 79, 63, 16, 227, 210, 56, 59, 14, 221, 201, 55, 196, 53, 48, 187, 11, 49, 191, 42, 180, 44, 175, 39, 154, 503, 168, 146, 37, 141, 491, 171, 132, 34, 137, 165, 478, 43, 128

Offset: 1

Antti Karttunen, Dec 10 2016

nonn

			a(1) = 1 and a(2) = 2 by definition.
Because A005187(2) = 3, a(3) = A055938(a(2)) = 5, and vice versa, as A055938(2) = 5, a(5) = A005187(a(2)) = 3.
Because A005187(3) = 4, a(4) = A055938(a(3)) = A055938(5) = 12.
Because A055938(3) = 6, a(6) = A005187(a(3)) = A005187(5) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Cf. A005187, A055938, A079559, A256992.

Related or similar permutations: A279341, A279342, A279343, A279344.

(definec (A279347 n) (cond ((<= n 2) n) ((zero? (A079559 n)) (A005187 (A279347 (A256992 n)))) (else (A055938 (A279347 (A256992 n))))))

a(1) = 1, a(2) = 2, for n > 2, A079559(n) = 0 [when n is a term of A055938], a(n) = A005187(a(A256992(n))), otherwise a(n) = A055938(a(A256992(n))).
As a composition of other permutations:
a(n) = A279342(A279343(n)).
a(n) = A279344(A279341(n)).

A323247 a(n) = A005187(A156552(n)).

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 15, 11, 10, 16, 31, 19, 63, 32, 18, 26, 127, 23, 255, 35, 34, 64, 511, 42, 22, 128, 25, 67, 1023, 39, 2047, 57, 66, 256, 38, 50, 4095, 512, 130, 74, 8191, 71, 16383, 131, 41, 1024, 32767, 89, 46, 47, 258, 259, 65535, 54, 70, 138, 514, 2048, 131071, 82, 262143, 4096, 73, 120, 134, 135, 524287, 515

Offset: 1

Antti Karttunen, Jan 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A001222, A005187, A006530, A156552, A252464, A253560, A323243, A323248.

Cf. also A283475.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
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))));
A323247(n) = A005187(A156552(n));

a(n) = A005187(A156552(n)).
a(n) = A323243(n) + A323248(n).
For n >= 2, a(A253560(n)) = a(n*A006530(n)) = a(n) + A000225(A001222(n)).
For n >= 1 and k >= 1, a(n*A000040(k+A000720(A006530(n)))) = a(n) + A000225(k+A001222(n)).

A324287 a(n) = A002487(A005187(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 3, 5, 4, 1, 4, 7, 5, 7, 7, 5, 5, 1, 5, 9, 7, 10, 11, 8, 7, 9, 9, 7, 13, 8, 3, 10, 6, 1, 6, 11, 9, 13, 15, 11, 10, 13, 14, 11, 21, 13, 5, 17, 9, 11, 11, 9, 19, 12, 5, 18, 19, 11, 3, 13, 7, 18, 15, 4, 7, 1, 7, 13, 11, 16, 19, 14, 13, 17, 19, 15, 29, 18, 7, 24, 13, 16, 17, 14, 30, 19, 8, 29, 31, 18, 5, 22, 12, 31, 26, 7

Offset: 0

Antti Karttunen, Feb 20 2019

nonn

The motivation for this kind of sequence was a question: what kind of simply defined non-injective functions f exist such that this sequence can be defined as their function, e.g., as a(n) = g(f(n)), where g is a nontrivial integer-valued function? The same question can also be asked about A324288, A324337 and A324338. Note that A005187, A283477 and A006068 used in their definitions are all injections. Of course, A324377(n) = A000265(A005187(n)) fills the bill as A002487(n) = A002487(A000265(n)), but are there any less obvious solutions? - Antti Karttunen, Feb 28 2019

Antti Karttunen, Table of n, a(n) for n = 0..16385
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A000265, A002487, A005187, A006068, A283477, A324286, A324288, A324293, A324337, A324338, A324377.

A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ Modified from the one given in A002487, sign not actually needed here.
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A324287(n) = A002487(A005187(n));

from functools import reduce
def A324287(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin((n<<1)-n.bit_count())[-1:2:-1],(1,0))) if n else 0 # Chai Wah Wu, May 05 2023

a(n) = A002487(A005187(n)).
a(n) = A324286(A283477(n)).
a(n) = A002487(A324377(n)).

A324288 a(n) = A002487(1+A005187(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 5, 2, 1, 5, 7, 3, 7, 2, 5, 8, 1, 6, 9, 4, 10, 3, 8, 13, 9, 2, 7, 12, 8, 11, 10, 7, 1, 7, 11, 5, 13, 4, 11, 18, 13, 3, 11, 19, 13, 18, 17, 12, 11, 2, 9, 16, 12, 17, 18, 13, 11, 14, 13, 10, 18, 11, 4, 13, 1, 8, 13, 6, 16, 5, 14, 23, 17, 4, 15, 26, 18, 25, 24, 17, 16, 3, 14, 25, 19, 27, 29, 21, 18, 23, 22, 17, 31, 19, 7, 23, 13, 2

Offset: 0

Antti Karttunen, Feb 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A002487, A005187, A324287, A324116.

A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ Modified from the one given in A002487, sign not actually needed here.
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A324288(n) = A002487(1+A005187(n));

from functools import reduce
def A324288(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(1+(n<<1)-n.bit_count())[-1:2:-1],(1,0))) if n else 1 # Chai Wah Wu, May 05 2023

a(n) = A002487(1+A005187(n)).

A280700 Binary weight of terms of A005187: a(n) = A000120(A005187(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 16 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A000120, A001221, A001222, A005187, A279357, A280705, A283475.

Table[DigitCount[2 n - DigitCount[2 n, 2, 1], 2, 1], {n, 0, 120}] (* Michael De Vlieger, Mar 18 2017 *)

b(n) = if(n<1, 0, b(n\2) + n%2);
for(n=0, 150, print1(b(2*n - b(2*n)), ", ")) \\ Indranil Ghosh, Mar 21 2017

def A(n): return bin(2*n - bin(2*n)[2:].count("1"))[2:].count("1")
print([A(n) for n in range(151)]) # Indranil Ghosh, Mar 21 2017

(define (A280700 n) (A000120 (A005187 n)))

a(n) = A000120(A005187(n)).

a(n) = A001221(A283475(n)) = A001222(A283475(n)) = A001222(A280705(n)).

A289617 a(n) = A005187(A001222(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 3, 7, 1, 4, 1, 4, 3, 3, 1, 7, 3, 3, 4, 4, 1, 4, 1, 8, 3, 3, 3, 7, 1, 3, 3, 7, 1, 4, 1, 4, 4, 3, 1, 8, 3, 4, 3, 4, 1, 7, 3, 7, 3, 3, 1, 7, 1, 3, 4, 10, 3, 4, 1, 4, 3, 4, 1, 8, 1, 3, 4, 4, 3, 4, 1, 8, 7, 3, 1, 7, 3, 3, 3, 7, 1, 7, 3, 4, 3, 3, 3, 10, 1, 4, 4, 7, 1, 4, 1, 7, 4, 3, 1, 8, 1, 4, 3, 8, 1, 4, 3, 4, 4, 3, 3, 8

Offset: 1

Antti Karttunen, Jul 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A001222, A005187, A289618.

Cf. A268375 (positions where coincides with A046645).

a(n) = my(bn=bigomega(n)); 2*bn-hammingweight(bn); \\ Michel Marcus, Jul 10 2017

(define (A289617 n) (A005187 (A001222 n)))

a(n) = A005187(A001222(n)).

cp's OEIS Frontend

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

A300244 Difference between A005187 and its Möbius transform (A297111).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A279342 a(0) = 1, a(1) = 2, a(2n) = A055938(a(n)), a(2n+1) = A005187(a(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A279344 a(0) = 1, a(2n) = A005187(a(n)), a(2n+1) = A055938(a(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A279347 Self-inverse permutation of natural numbers: a(1) = 1, a(2) = 2, a(A055938(n)) = A005187(a(n)), a(A005187(n)) = A055938(a(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

Formula

A323247 a(n) = A005187(A156552(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A324287 a(n) = A002487(A005187(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Formula

A324288 a(n) = A002487(1+A005187(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python