A065620 -id:A065620 - cp's OEIS frontend

A325565 a(n) is the number of such divisors d of n that A048720(A065621(d),n/d) is equal to n.

1, 2, 1, 3, 1, 2, 1, 4, 2, 2, 1, 3, 1, 2, 1, 5, 1, 4, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 2, 2, 2, 6, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 5, 2, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 2, 4, 1, 3, 1, 4, 1, 8, 1, 2, 2, 3, 1, 2, 1, 5, 1, 2, 1, 6, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 6, 1, 4, 1, 3, 1, 2, 1, 4, 2

Offset: 1

Antti Karttunen, May 09 2019

nonn

Equally, a(n) is number of such pairs of natural numbers t, u that A048720(t,u) = n and A065620(t)*u = n.

Cf. A000005, A000265, A001511, A048720, A065620, A065621, A091220, A115872, A325566, A325567, A325568, A325569, A325570 (positions of ones).

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A325565(n) = sumdiv(n,d,A048720(A065621(d),n/d)==n);

A065620(n, c=1) = sum(i=0, logint(n+!n, 2), if(bittest(n, i), (-1)^c++<A065620
A325565(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sum(d=1,n,my(q = Pol(binary(d))*Mod(1, 2)); (0==(p%q) && (n==(A065620(d)*fromdigits(Vec(lift(p/q)),2))))); };

a(n) = Sum_{d|n} [A048720(A065621(d),n/d) == n], where [ ] is the Iverson bracket.
a(n) / a(A000265(n)) = A001511(n).
a(n) <= A000005(n) for all n.
a(n) <= A091220(n) for all n.

A348296 Irregular table T(n, k), n > 0, k = 1..A000120(n), read by rows; the n-th contains, in ascending order, the distinct powers of 2 summing to n.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 4, 2, 4, 1, 2, 4, 8, 1, 8, 2, 8, 1, 2, 8, 4, 8, 1, 4, 8, 2, 4, 8, 1, 2, 4, 8, 16, 1, 16, 2, 16, 1, 2, 16, 4, 16, 1, 4, 16, 2, 4, 16, 1, 2, 4, 16, 8, 16, 1, 8, 16, 2, 8, 16, 1, 2, 8, 16, 4, 8, 16, 1, 4, 8, 16, 2, 4, 8, 16, 1, 2, 4, 8, 16, 32

Offset: 1

Rémy Sigrist, Jul 18 2022

			Triangle T(n, k) begins:
  n   n-th row
  --  ------------
   1  [1]
   2  [2]
   3  [1, 2]
   4  [4]
   5  [1, 4]
   6  [2, 4]
   7  [1, 2, 4]
   8  [8]
   9  [1, 8]
  10  [2, 8]
  11  [1, 2, 8]
  12  [4, 8]
  13  [1, 4, 8]
  14  [2, 4, 8]
  15  [1, 2, 4, 8]

Michael De Vlieger, Table of n, a(n) for n = 1..10870 (rows n = 1..2000, flattened)
Index entries for sequences related to binary expansion of n

Cf. A000079, A000120, A006519, A053644, A059867, A065620, A133457.

Array[DeleteCases[Union@ NumberExpand[#, 2], 0] &, 32] // Flatten (* Michael De Vlieger, Jul 19 2022 *)

row(n) = { my (r=vector(hammingweight(n))); for (k=1, #r, n -= r[k] = 2^valuation(n, 2)); return (r) }

T(n, k) = 2^A133457(n, k).
T(n, 1) = A006519(n).
T(n, A000120(n)) = A053644(n).
Sum_{k = 1..A000120(n)} T(n, k) = n.
Sum_{k = 1..A000120(n)} T(n, k) * (-1)^(k-1) = A065620(n).
Product_{k = 1..A000120(n)} T(n, k) = A059867(n).
T(2*n, k) = 2*T(n, k).

A246164 Permutation of natural numbers: a(1) = 1, a(A065621(n)) = A014580(a(n-1)), a(A048724(n)) = A091242(a(n)), where A065621(n) and A048724(n) are the reversing binary representation of n and -n, respectively, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

Original entry on oeis.org

1, 2, 4, 11, 8, 5, 3, 7, 6, 9, 13, 17, 47, 31, 14, 61, 21, 42, 185, 24, 87, 319, 62, 12, 25, 19, 10, 59, 20, 15, 37, 229, 49, 22, 67, 76, 415, 103, 28, 18, 55, 137, 34, 41, 16, 27, 97, 78, 425, 109, 29, 1627, 222, 54, 283, 433, 79, 373, 3053, 33, 131, 647, 108, 847, 133, 745, 6943, 44, 193, 1053, 160, 504, 4333, 587, 99

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where the two complementary pairs to be entangled with each other are A065621/A048724 and A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)).
The former are themselves permutations of A000069/A001969 (odious and evil numbers), which means that this permutation shares many properties with A246162.
For the comments about the cycle structure, please see A246163.

Inverse: A246163.
Similar or related permutations: A246206, A246202, A193231, A245702, A234025, A246162, A234612, A246204.
Cf. A010060, A014580, A048724, A065620, A065621, A091242, A246159, A246160.

a(1) = 1, and for n > 1, if A010060(n) = 1 [i.e. when n is an odious number], a(n) = A014580(a(A065620(n)-1)), otherwise a(n) = A091242(a(- (A065620(n)))). [A065620 Converts sum of powers of 2 in binary representation of n to an alternating sum].
As a composition of related permutations:
a(n) = A246202(A193231(n)).
a(n) = A245702(A234025(n)).
a(n) = A246162(A234612(n)).
a(n) = A193231(A246204(A193231(n))).
For all n > 1, A091225(a(n)) = A010060(n). [Maps odious numbers to binary representations of irreducible GF(2) polynomials (A014580) and evil numbers to the corresponding representations of reducible polynomials (A091242), in some order. A246162 has the same property].

A246159 Inverse function to the injection A048724.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 2, 0, 0, 7, 6, 0, 4, 0, 0, 5, 0, 15, 14, 0, 12, 0, 0, 13, 8, 0, 0, 9, 0, 11, 10, 0, 0, 31, 30, 0, 28, 0, 0, 29, 24, 0, 0, 25, 0, 27, 26, 0, 16, 0, 0, 17, 0, 19, 18, 0, 0, 23, 22, 0, 20, 0, 0, 21, 0, 63, 62, 0, 60, 0, 0, 61, 56, 0, 0, 57, 0, 59, 58, 0, 48, 0, 0, 49, 0, 51, 50, 0, 0, 55, 54, 0, 52, 0, 0, 53, 32

Offset: 0

Antti Karttunen, Aug 18 2014

nonn

After a(0)=0, sequence has nonzero values a(n) = k at those positions n for which A048724(k) = n and zeros at those positions n which are not present in A048724.

Equally, sequence is obtained when the positive terms of A065620 are replaced with zeros and the sign of the negative terms is reversed.

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

Cf. A246160, A006068, A010059, A048724, A065620, A245710.

a065620(n) = if(n<3, n, if(n%2, -2*a065620((n - 1)/2) + 1, 2*a065620(n/2)));
a(n) = -!(hammingweight(n)%2)*a065620(n);
for(n=0, 100, print1(a(n),", ")) \\ Indranil Ghosh, Jun 07 2017

def a065620(n): return n if n<3 else 2*a065620(n//2) if n%2==0 else -2*a065620((n - 1)//2) + 1
def a(n): return -(bin(n)[2:].count("1")%2==0)*a065620(n)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(n) = (1/2) * A010059(n) * A006068(n).
a(n) = -1 * A010059(n) * A065620(n).
a(n) = A246160(n) - A065620(n).
a(n) = A010059(n) * A006068(A245710(n)).
For all n, a(A048724(n)) = n.

A246160 Inverse function to the injection A065621.

Original entry on oeis.org

0, 1, 2, 0, 4, 0, 0, 3, 8, 0, 0, 7, 0, 5, 6, 0, 16, 0, 0, 15, 0, 13, 14, 0, 0, 9, 10, 0, 12, 0, 0, 11, 32, 0, 0, 31, 0, 29, 30, 0, 0, 25, 26, 0, 28, 0, 0, 27, 0, 17, 18, 0, 20, 0, 0, 19, 24, 0, 0, 23, 0, 21, 22, 0, 64, 0, 0, 63, 0, 61, 62, 0, 0, 57, 58, 0, 60, 0, 0, 59, 0, 49, 50, 0, 52

Offset: 0

Antti Karttunen, Aug 18 2014

nonn

Sequence has nonzero values a(n) = k at those points n for which A065621(k) = n and zeros at those positions n which are not present in A065621.

Equally, sequence is obtained when the negative terms of A065620 are replaced with zeros

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A246159, A006068, A010060, A115384, A065620, A065621.

a065620(n) = if(n<3, n, if(n%2, -2*a065620((n - 1)/2) + 1, 2*a065620(n/2)));
a(n) = (hammingweight(n)%2)*a065620(n);
for(n=0, 100, print1(a(n),", ")) \\ Indranil Ghosh, Jun 07 2017

def a065620(n): return n if n<3 else 2*a065620(n//2) if n%2==0 else -2*a065620((n - 1)//2) + 1
def a(n): return (bin(n)[2:].count("1")%2)*a065620(n)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(n) = A010060(n) * A065620(n).
a(n) = A246159(n) + A065620(n).
a(0) = 0, and for n >= 1, a(n) = A010060(n) *  (1 + A006068(A115384(n)-1)).
For all n, a(A065621(n)) = n.

A273672 Permutation of natural numbers induced by looking up the position of fraction A270418(n)/A270419(n) from the full Stern-Brocot tree A007305(n+1)/A047679(n-1).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 2, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 6, 33554431, 67108863, 4, 268435455, 536870911, 1073741823, 2147483647, 128, 8589934591, 17179869183, 34359738367, 68719476735, 137438953471, 274877906943, 549755813887, 14

Offset: 1

Antti Karttunen, May 27 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A273671.
Cf. A270418, A270419.
Cf. also A007305, A047679.

A065620(n, c=1) = sum(i=0, logint(n+!n, 2), if(bittest(n, i), (-1)^c++<A065620
SBtree_index(r) = { my(m=numerator(r),n=denominator(r),z=1); while(m!=n, if(mA273672(n) = { n=factor(n); n[, 2] = apply(A065620, n[, 2]); SBtree_index(factorback(n)); }; \\ Antti Karttunen, Mar 07 2020, based also on M. F. Hasler's code in A270418 and A270419

(define (A273672 n) (SBtree_index (A270418 n) (A270419 n)))
(define (SBtree_index m n) (let loop ((m m) (n n) (z 1)) (cond ((= m n) z) ((< m n) (loop m (- n m) (+ z z))) (else (loop (- m n) n (+ z z 1))))))

A295881 Reversing binary representation of the deficiency of n, A033879(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 14, 1, 13, 2, 26, 12, 28, 4, 14, 1, 16, 5, 50, 6, 26, 8, 62, 20, 55, 26, 22, 0, 44, 20, 38, 1, 50, 22, 62, 53, 100, 16, 62, 30, 104, 20, 122, 4, 28, 52, 118, 36, 121, 11, 38, 14, 84, 20, 110, 24, 98, 42, 74, 80, 76, 44, 62, 1, 118, 20, 194, 26, 122, 12, 206, 85, 200, 98, 42, 28, 74, 20, 214, 46, 121

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

For all n, A010060(a(A005100(n))) = 1 and A010060(a(A023196(n))) = 0. That is, for the deficient numbers a(n) is an odious number (A000069) and for the nondeficient numbers a(n) is an evil number (A001969).

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

Cf. A005100, A023196, A033879, A033880, A048724, A065620, A065621, A286449, A295882.

Cf. A000396 (gives the positions of zeros).

(define (A295881 n) (let ((x (A033879 n))) (if (<= x 0) (A048724 (- x)) (A065621 x))))

If A033879(n) <= 0, a(n) = A048724(-A033879(n)), otherwise a(n) = A065621(A033879(n)).

For all n >= 1, A065620(a(n)) = A033879(n).

A104895 a(0)=0; thereafter a(2n) = -2a(n), a(2n+1) = 2a(n) - 1.

Original entry on oeis.org

0, -1, -2, 1, -4, 3, 2, -3, -8, 7, 6, -7, 4, -5, -6, 5, -16, 15, 14, -15, 12, -13, -14, 13, 8, -9, -10, 9, -12, 11, 10, -11, -32, 31, 30, -31, 28, -29, -30, 29, 24, -25, -26, 25, -28, 27, 26, -27, 16, -17, -18, 17, -20, 19, 18, -19, -24, 23, 22, -23, 20, -21, -22, 21, -64, 63, 62, -63, 60, -61, -62, 61, 56, -57, -58, 57, -60, 59

Offset: 0

Philippe Deléham, Apr 24 2005

Columns of table in A104894 written in base 10.

Conjecture: the positions where 0, 1, 2, 3, ... appear are given by A048724; the positions where -1, -2, -3, ... appear are given by A065621.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Cf. A048724, A065621, A104894.

The negative of entry A065620.

import Data.List (transpose)
a104895 n = a104895_list !! n
a104895_list = 0 : concat (transpose [map (negate . (+ 1)) zs, tail zs])
               where zs = map (* 2) a104895_list
-- Reinhard Zumkeller, Mar 26 2014

f:=proc(n) option remember; if n=0 then RETURN(0); fi; if n mod 2 = 0 then RETURN(2*f(n/2)); else RETURN(-2*f((n-1)/2)-1); fi; end;

a[0] = 0;
a[n_]:= a[n]= If[EvenQ[n], 2 a[n/2], -2 a[(n-1)/2] - 1];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 03 2018 *)

def a(n):
    if (n==0): return 0
    elif (mod(n,2)==0): return 2*a(n/2)
    else: return -2*a((n-1)/2) - 1
[a(n) for n in (0..100)] # G. C. Greubel, Jun 15 2021

a(0) = 0 and for k>=0, 0<= j <2^k, a(2^k + j) = a(j) + 2^k if a(j)<0, a(2^k + j) = a(j) - 2^k if a(j)>=0.
Sum_{0 <= n <= 2^k - 1} a(n) = - 2^(k-1).
Sum_{0 <= n <= 2^k - 1} |a(n)| = 4^(k-1).
a(n) = -A065620(n). - M. F. Hasler, Apr 16 2018

Corrected by N. J. A. Sloane, Nov 05 2005

Edited by N. J. A. Sloane, Apr 25 2018

A323908 Reversing binary representation of A004718, Per Nørgård's "infinity sequence".

Original entry on oeis.org

0, 1, 3, 2, 1, 0, 6, 7, 3, 2, 0, 1, 2, 3, 5, 4, 1, 0, 6, 7, 0, 1, 3, 2, 6, 7, 1, 0, 7, 6, 12, 13, 3, 2, 0, 1, 2, 3, 5, 4, 0, 1, 3, 2, 1, 0, 6, 7, 2, 3, 5, 4, 3, 2, 0, 1, 5, 4, 2, 3, 4, 5, 15, 14, 1, 0, 6, 7, 0, 1, 3, 2, 6, 7, 1, 0, 7, 6, 12, 13, 0, 1, 3, 2, 1, 0, 6, 7, 3, 2, 0, 1, 2, 3, 5, 4, 6, 7, 1, 0, 7, 6, 12, 13, 1, 0

Offset: 0

Antti Karttunen, Feb 09 2019

nonn

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

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

Cf. A004718, A048724, A065620, A065621, A083866 (positions of zeros), A323907 (rgs-transform), A323909.

up_to = 65536;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ After code in A004718.
v004718 = A004718list(up_to);
A004718(n) = if(!n,n,v004718[n]);
A048724(n) = bitxor(n, n<<1);
A065621(n) = bitxor(n-1,n+n-1);
A323908(n) = if(A004718(n)<=0, A048724(-A004718(n)), A065621(A004718(n)));

from itertools import groupby
def A323908(n):
    c = 0
    for k, g in groupby(bin(n)[2:]):
        c = c+len(list(g)) if k == '1' else (-c if len(list(g))&1 else c)
    return -c^(-c<<1) if c<=0 else c^(c&~-c)<<1 # Chai Wah Wu, Mar 02 2023

If A004718(n) <= 0, a(n) = A048724(-A004718(n)), otherwise a(n) = A065621(A004718(n)).

For all n >= 1, A065620(a(n)) = A004718(n).

A297153 Reversing binary representation of A083254(n), 2*phi(n) - n.

Original entry on oeis.org

1, 0, 1, 0, 7, 6, 13, 0, 7, 6, 25, 12, 31, 6, 1, 0, 19, 10, 49, 12, 7, 6, 61, 24, 19, 6, 25, 12, 47, 18, 37, 0, 11, 6, 21, 20, 103, 6, 25, 24, 107, 54, 121, 12, 7, 6, 117, 48, 103, 30, 21, 12, 87, 54, 41, 24, 19, 6, 73, 36, 79, 6, 25, 0, 35, 46, 193, 12, 55, 58, 205, 40, 203, 6, 13, 12, 127, 34, 213, 48, 47, 6, 241

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

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

Cf. A000010, A048724, A065620, A065621, A083254.

Cf. also A297154, A295881.

(define (A297153 n) (let ((x (A083254 n))) (if (> x 0) (A065621 x) (A048724 (- x)))))

If A083254(n) > 0, then a(n) = A065621(A083254(n)), otherwise a(n) = A048724(-A083254(n)).

For all n >= 1, A065620(a(n)) = A083254(n)

cp's OEIS Frontend

A325565 a(n) is the number of such divisors d of n that A048720(A065621(d),n/d) is equal to n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A348296 Irregular table T(n, k), n > 0, k = 1..A000120(n), read by rows; the n-th contains, in ascending order, the distinct powers of 2 summing to n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A246159 Inverse function to the injection A048724.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Formula

A246160 Inverse function to the injection A065621.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Formula

A273672 Permutation of natural numbers induced by looking up the position of fraction A270418(n)/A270419(n) from the full Stern-Brocot tree A007305(n+1)/A047679(n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Scheme

A295881 Reversing binary representation of the deficiency of n, A033879(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A104895 a(0)=0; thereafter a(2n) = -2*a(n), a(2n+1) = 2*a(n) - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A104895 a(0)=0; thereafter a(2n) = -2a(n), a(2n+1) = 2a(n) - 1.