A088161 -id:A088161 - cp's OEIS frontend

A088163 Numbers for which rotating one binary place to the right less rotating one binary place to the left is equal to zero.

0, 1, 2, 3, 7, 10, 15, 31, 42, 63, 127, 170, 255, 511, 682, 1023, 2047, 2730, 4095, 8191, 10922, 16383, 32767, 43690, 65535, 131071, 174762, 262143, 524287, 699050, 1048575, 2097151, 2796202, 4194303, 8388607, 11184810, 16777215, 33554431, 44739242, 67108863

Offset: 0

Robert G. Wilson v, Sep 13 2003

n is a member iff n is of the form 2^n -1 (A000225) or A000975(2n).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,5,0,0,-4).

Cf. A000225, A000975.

f[n_] := FromDigits[ RotateRight[ IntegerDigits[n, 2]], 2] - FromDigits[ RotateLeft[ IntegerDigits[n, 2]], 2]; Select[ Range[33560000], f[ # ] == 0 &]
(* Or *) Union[ Join[ Table[2^n - 1, {n, 0, 25}], Table[ Ceiling[2(2^n - 1)/3], {n, 2, 24, 2}]]]
LinearRecurrence[{0,0,5,0,0,-4},{0,1,2,3,7,10},40] (* Harvey P. Dale, Feb 20 2022 *)

concat(0, Vec(x*(1+x)*(1+x+2*x^2) / ((1-x)*(1+x+x^2)*(1-4*x^3)) + O(x^50))) \\ Colin Barker, May 14 2016

Numbers n such that A038572(n) - A006257(n) = A088161(n) = 0.
From Colin Barker, May 14 2016: (Start)
a(n) = 5*a(n-3)-4*a(n-6) for n>5.
G.f.: x*(1+x)*(1+x+2*x^2) / ((1-x)*(1+x+x^2)*(1-4*x^3)).
(End)

A273105 a(n) = A038572(n) + A006257(n), sum of the two numbers obtained by rotating the binary representation of n by one place to the right and to the left.

Original entry on oeis.org

0, 2, 2, 6, 3, 9, 8, 14, 5, 15, 10, 20, 15, 25, 20, 30, 9, 27, 14, 32, 19, 37, 24, 42, 29, 47, 34, 52, 39, 57, 44, 62, 17, 51, 22, 56, 27, 61, 32, 66, 37, 71, 42, 76, 47, 81, 52, 86, 57, 91, 62, 96, 67, 101, 72, 106, 77, 111, 82, 116, 87, 121, 92, 126, 33, 99

Offset: 0

Alex Ratushnyak, May 15 2016

Cf. A006257, A038572, A088161, A088163.

Table[FromDigits[RotateRight@ #, 2] + FromDigits[RotateLeft@ #, 2] &@ IntegerDigits[n, 2], {n, 0, 65}] (* Michael De Vlieger, May 17 2016 *)

print('0', end=',')
for n in range(1,1000):
    BL = len(bin(n))-2
    x = (n>>1) + ((n&1) << (BL-1))   # A038572(n)
    x+= (n*2) - (1<A006257(n)  for n>0
    print(str(x), end=',')

A273106 Numbers representable as ror(k)+rol(k), where ror(k)=A038572(k) is k rotated one binary place to the right, rol(k)=A006257(k) is k rotated one binary place to the left.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 9, 10, 14, 15, 17, 19, 20, 22, 24, 25, 27, 29, 30, 32, 33, 34, 37, 38, 39, 42, 43, 44, 47, 48, 51, 52, 53, 56, 57, 58, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 91, 92, 93, 95, 96, 98

Offset: 0

Alex Ratushnyak, May 15 2016

Cf. A006257, A038572, A088161, A088163, A273105.

Take[#, 66] &@ Union@ Table[FromDigits[RotateRight@ #, 2] + FromDigits[RotateLeft@ #, 2] &@ IntegerDigits[n, 2], {n, 0, 10^3}] (* Michael De Vlieger, May 17 2016 *)

def ROR(n):                # returns A038572(n)
    BL = len(bin(n))-2
    return (n>>1) + ((n&1) << (BL-1))
def ROL(n):                # returns A006257(n) for n>0
    BL = len(bin(n))-2
    return (n*2) - (1<

A273180 Numbers n such that ror(n) + rol(n) is a power of 2, where ror(n)=A038572(n) is n rotated one binary place to the right, rol(n)=A006257(n) is n rotated one binary place to the left.

Original entry on oeis.org

1, 2, 6, 19, 38, 102, 307, 614, 1638, 4915, 9830, 26214, 78643, 157286, 419430, 1258291, 2516582, 6710886, 20132659, 40265318, 107374182, 322122547, 644245094, 1717986918, 5153960755, 10307921510, 27487790694, 82463372083, 164926744166, 439804651110

Offset: 1

Alex Ratushnyak, May 17 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,17,0,0,-16).

Cf. A006257, A038572, A088161, A088163, A112627, A273105, A273106.

#include 
int main(int argc, char** argv)
{
  unsigned long long x, n, BL=0;
  for (n=1; n>0; ++n) {
    if ((n & (n-1))==0)  ++BL;
    x = (n>>1) + ((n&1) << (BL-1));   // A038572(n)
    x+= (n*2) - (1ull<A006257(n)  for n>0
    if ((x & (x-1))==0)  printf("%lld, ", n);
  }
}

Select[Range[10^6], IntegerQ@ Log2[FromDigits[RotateRight@ #, 2] + FromDigits[RotateLeft@ #, 2]] &@ IntegerDigits[#, 2] &] (* or *)
Rest@ CoefficientList[Series[x (1 + 2 x + 6 x^2 + 2 x^3 + 4 x^4)/((1 - x) (1 + x + x^2) (1 - 16 x^3)), {x, 0, 30}], x] (* Michael De Vlieger, May 19 2016 *)

Vec(x*(1+2*x+6*x^2+2*x^3+4*x^4)/((1-x)*(1+x+x^2)*(1-16*x^3)) + O(x^50)) \\ Colin Barker, May 19 2016

From Colin Barker, May 19 2016: (Start)
a(n) = 17*a(n-3) - 16*a(n-6) for n>6.
G.f.: x*(1+2*x+6*x^2+2*x^3+4*x^4) / ((1-x)*(1+x+x^2)*(1-16*x^3)).
(End)

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A088163 Numbers for which rotating one binary place to the right less rotating one binary place to the left is equal to zero.

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A273105 a(n) = A038572(n) + A006257(n), sum of the two numbers obtained by rotating the binary representation of n by one place to the right and to the left.

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A273106 Numbers representable as ror(k)+rol(k), where ror(k)=A038572(k) is k rotated one binary place to the right, rol(k)=A006257(k) is k rotated one binary place to the left.

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A273180 Numbers n such that ror(n) + rol(n) is a power of 2, where ror(n)=A038572(n) is n rotated one binary place to the right, rol(n)=A006257(n) is n rotated one binary place to the left.

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