A006257 -id:A006257 - cp's OEIS frontend

A256249 Partial sums of A006257 (Josephus problem).

0, 1, 2, 5, 6, 9, 14, 21, 22, 25, 30, 37, 46, 57, 70, 85, 86, 89, 94, 101, 110, 121, 134, 149, 166, 185, 206, 229, 254, 281, 310, 341, 342, 345, 350, 357, 366, 377, 390, 405, 422, 441, 462, 485, 510, 537, 566, 597, 630, 665, 702, 741, 782, 825, 870, 917, 966, 1017, 1070, 1125, 1182, 1241, 1302, 1365, 1366, 1369, 1374

Offset: 0

Omar E. Pol, Mar 20 2015

Also total number of ON states after n generations in one of the four wedges of the one-step rook version (or in one of the four quadrants of the one-step bishop version) of the cellular automaton of A256250.
A006257 gives the number of cells turned ON at n-th stage.
First differs from A255747 at a(11).
First differs from A169779 at a(10).
It appears that the odd terms (a bisection) give A256250.

			Written as an irregular triangle T(n,k), k >= 1, in which the row lengths are the terms of A011782 the sequence begins:
   0;
   1;
   2,  5;
   6,  9, 14, 21;
  22, 25, 30, 37, 46, 57, 70, 85;
  86, 89, 94,101,110,121,134,149,166,185,206,229,254,281,310,341;
  ...
Right border, a(2^m-1), gives A002450(m) for m >= 0.
a(2^m-2) = A203241(m) for m >= 2.
It appears that this triangle at least shares with the triangles from the following sequences; A151920, A255737, A255747, the positive elements of the columns k, if k is a power of 2.
From _Omar E. Pol_, Jan 03 2016: (Start)
Illustration of initial terms in the fourth quadrant of the square grid:
---------------------------------------------------------------------------
n    a(n)                 Compact diagram
---------------------------------------------------------------------------
0     0     _
1     1    |_|_ _
2     2      |_| |
3     5      |_ _|_ _ _ _
4     6          |_| | | |
5     9          |_ _| | |
6    14          |_ _ _| |
7    21          |_ _ _ _|_ _ _ _ _ _ _ _
8    22                  |_| | | | | | | |
9    25                  |_ _| | | | | | |
10   30                  |_ _ _| | | | | |
11   37                  |_ _ _ _| | | | |
12   46                  |_ _ _ _ _| | | |
13   57                  |_ _ _ _ _ _| | |
14   70                  |_ _ _ _ _ _ _| |
15   85                  |_ _ _ _ _ _ _ _|
.
a(n) is also the total number of cells in the first n regions of the diagram. A006257(n) gives the number of cells in the n-th region of the diagram.
(End)

David A. Corneth, Table of n, a(n) for n = 0..9999
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 37, 41.
Yuri Nikolayevsky and Ioannis Tsartsaflis, Cohomology of N-graded Lie algebras of maximal class over Z_2, arXiv:1512.87676 [math.RA], (2016), page 6.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to the Josephus Problem

Cf. A002450, A006257, A011782, A139250, A151920, A169779, A255737, A255747, A256250, A256251, A266540.

(* Based on Birkas Gyorgy's code in A006257 *)
Accumulate[Prepend[Flatten[Table[Range[1,2^n-1,2],{n,0,7}]],0]] (* Ivan N. Ianakiev, Mar 30 2015 *)

a(n)=n++;b=#binary(n>>1);(4^b-1)/3+(n-2^b)^2 \\ David A. Corneth, Mar 21 2015

a(n) = (A256250(n+1) - 1)/4.

A266539 Terms of A006257 (Josephus problem) repeated.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 3, 3, 1, 1, 3, 3, 5, 5, 7, 7, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25

Offset: 1

Omar E. Pol, Jan 02 2016

First differs from both A266509 and A266529 at a(25), and shares with them infinitely many terms.

			Written as an irregular triangle in which the row lengths are twice the terms of A011782 the sequence begins:
   0, 0;
   1, 1;
   1, 1, 3, 3;
   1, 1, 3, 3, 5, 5, 7, 7;
   1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15;
   ...
Row sums give 0 together with A004171.

Partial sums give A266540.

Cf. A004171, A006257, A011782, A256249, A266509, A266529, A266535.

A006257[0]:=0: for n from 1 to 100 do A006257[n]:=(A006257[n-1]+1) mod n +1: end do:
seq(A006257[i]$2,i=0..100); # Robert Israel, Jan 13 2016

Join[{0, 0}, Table[SeriesCoefficient[(1 + x^2)/((-1 + x)^2 (1 + x)), {x, 0, m}], {n, 6}, {m, 0, 2^n - 1}]] // Flatten (* Michael De Vlieger, Jan 05 2016 *)

G.f.: (x^2 + x^4)/(1 - x - x^2 + x^3) - (1 - x)^(-1)*Sum_{k>=1} 2^k*x^(2^(k+1)). - Robert Israel, Jan 13 2016

Offset changed to 1 by Ivan Neretin, Feb 09 2017

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=',')

A266538 Twice the partial sums of A006257 (Josephus problem).

Original entry on oeis.org

0, 2, 4, 10, 12, 18, 28, 42, 44, 50, 60, 74, 92, 114, 140, 170, 172, 178, 188, 202, 220, 242, 268, 298, 332, 370, 412, 458, 508, 562, 620, 682, 684, 690, 700, 714, 732, 754, 780, 810, 844, 882, 924, 970, 1020, 1074, 1132, 1194, 1260, 1330, 1404, 1482, 1564, 1650, 1740, 1834, 1932, 2034, 2140, 2250, 2364, 2482, 2604

Offset: 0

Omar E. Pol, Jan 12 2016

Index entries for sequences related to the Josephus Problem.

Bisection of A266540.

Cf. A006257, A256249, A266535, A266540.

2*Accumulate[Flatten[{0, Table[Range[1, 2^n - 1, 2], {n, 0, 7}]}]] (* Jake L Lande, Aug 05 2024 *)

a(n) = 2 * A256249(n).

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)

A274524 Numbers n such that both ror(n) and rol(n) are squares, where ror(x)=A038572(x) is x rotated one binary place to the right, rol(x)=A006257(x) is x rotated one binary place to the left.

Original entry on oeis.org

1, 2, 8, 32, 128, 512, 1568, 2048, 2312, 2592, 2888, 8192, 16928, 32768, 131072, 139392, 250632, 524288, 549152, 566048, 672800, 924800, 963272, 1318688, 2097152, 8388608, 8520192, 8769672, 9005768, 12261152, 13582472, 15635232, 33554432, 134217728, 136059008, 136587392

Offset: 1

Alex Ratushnyak, Jun 27 2016

A004171 and A081294 are subsequences.
From Robert Israel, Jul 13 2016: (Start)
All terms except 1 are even.
Even terms are the numbers of the form n = (a+b)^2/8 such that for some d >= 1,
2^d <= n < 2^(d+1) and 2^(d+1)-1 = a*b. (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006257, A038572, A004171, A081294.

F:= proc(d) local v,R,X;
      v:= 2^(d+1)-1;
      R:= select(t-> t^2 < v,numtheory:-divisors(v));
      op(select(t -> t >= (v+1)/2 and t < v+1, map(t -> (t+ v/t)^2/8, R)));
end proc:
sort(convert({1,seq(F(i),i=1..50)},list)); # Robert Israel, Jul 13 2016

Select[Range[10^6], Times @ Boole@ {IntegerQ@ Sqrt@ FromDigits[RotateRight@ #, 2], IntegerQ@ Sqrt@ FromDigits[RotateLeft@ #, 2]} &@ IntegerDigits[#, 2] == 1 &] (* Michael De Vlieger, Jun 29 2016 *)

A273050 Numbers k such that ror(k) XOR rol(k) = k, where ror(x)=A038572(x) is x rotated one binary place to the right, rol(x)=A006257(x) is x rotated one binary place to the left, and XOR is the binary exclusive-or operator.

Original entry on oeis.org

0, 5, 6, 45, 54, 365, 438, 2925, 3510, 23405, 28086, 187245, 224694, 1497965, 1797558, 11983725, 14380470, 95869805, 115043766, 766958445, 920350134, 6135667565, 7362801078, 49085340525, 58902408630, 392682724205, 471219269046

Offset: 1

Alex Ratushnyak, May 13 2016

Cf. A006257, A038572, A088163, A125836 (bisection?), A125837 (bisection?).

Cf. A020988 (numbers k such that ror(k) + rol(k) = k).

ok[n_] := Block[{x = IntegerDigits[n, 2]}, x == BitXor @@@ Transpose@ {RotateLeft@ x, RotateRight@ x}]; Select[ Range[0, 10^5], ok] (* Giovanni Resta, May 14 2016 *)
ok[n_] := Block[{x = IntegerDigits[n, 2]}, x == BitXor @@@ Transpose[ {RotateLeft[x], RotateRight[x]}]]; Select[LinearRecurrence[{0, 9, 0, -8}, {0, 5, 6, 45}, 100], ok] (* Jean-François Alcover, May 22 2016, after Giovanni Resta *)

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<

Conjectures from Colin Barker, May 22 2016: (Start)
a(n) = (-11+(-1)^n+2^(-1/2+(3*n)/2)*(3-3*(-1)^n+5*sqrt(2)+5*(-1)^n*sqrt(2)))/14.
a(n) = 5*(2^(3*n/2)-1)/7 for n even.
a(n) = 3*(2^((3*n)/2-1/2)-2)/7 for n odd.
a(n) = 9*a(n-2)-8*a(n-4) for n>4.
G.f.: x^2*(5+6*x) / ((1-x)*(1+x)*(1-8*x^2)).
(End)

a(19)-a(27) from Giovanni Resta, May 14 2016

A274133 Numbers representable as ror(x)+rol(x) in two or more ways, where ror(x)=A038572(x) is x rotated one binary place to the right, rol(x)=A006257(x) is x rotated one binary place to the left.

Original entry on oeis.org

2, 9, 14, 15, 20, 27, 32, 37, 42, 47, 52, 57, 62, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 184, 189, 194, 195, 199, 200, 204, 205, 209, 210, 214, 215, 219, 220, 224, 225, 229, 230, 234, 235, 239, 240, 244, 245, 249, 250, 254, 255, 260, 265, 270, 275

Offset: 1

Alex Ratushnyak, Jun 10 2016

These are the duplicates in A273105.

			ror(5) + rol(5) = 6 + 3 = 9, and also ror(16) + rol(16) = 8 + 1 = 9, therefore 9 is in the sequence.

Cf. A006257, A038572, A273105.

A274134 Primes p such that both ror(p) and rol(p) are also primes, where ror(x)=A038572(x) is x rotated one binary place to the right, rol(x)=A006257(x) is x rotated one binary place to the left.

Original entry on oeis.org

3, 7, 11, 31, 43, 67, 79, 127, 131, 139, 167, 191, 211, 223, 227, 307, 331, 367, 487, 523, 631, 691, 743, 751, 883, 971, 1039, 1087, 1399, 2063, 2083, 2143, 2179, 2239, 2267, 2287, 2347, 2411, 2423, 2503, 2531, 2543, 2591, 2687, 2731, 2803, 2819, 2927, 2939, 2963

Offset: 1

Alex Ratushnyak, Jun 10 2016

a(n) mod 4 = 3.

Cf. A000040, A006257, A038572.

Select[Prime@ Range@ 430, And[PrimeQ@ FromDigits[RotateLeft@ #, 2], PrimeQ@ FromDigits[RotateRight@ #, 2]] &@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Jun 22 2016 *)

from sympy import isprime
for n in range(3, 10000, 2):
    if not isprime(n): continue
    BL = len(bin(n))-2
    x = (n>>1) + ((n&1) << (BL-1))   # A038572(n)
    if not isprime(x): continue
    y = (n*2) - (1<A006257(n)  for n>0
    if not isprime(y): continue
    print(str(n), end=', ')

cp's OEIS Frontend

A256249 Partial sums of A006257 (Josephus problem).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A266539 Terms of A006257 (Josephus problem) repeated.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Python

A266538 Twice the partial sums of A006257 (Josephus problem).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Python

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.

Views

Author

Keywords

Links

Crossrefs

Programs

C

Mathematica

PARI

Formula

A274524 Numbers n such that both ror(n) and rol(n) are squares, where ror(x)=A038572(x) is x rotated one binary place to the right, rol(x)=A006257(x) is x rotated one binary place to the left.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A273050 Numbers k such that ror(k) XOR rol(k) = k, where ror(x)=A038572(x) is x rotated one binary place to the right, rol(x)=A006257(x) is x rotated one binary place to the left, and XOR is the binary exclusive-or operator.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Python