A003188 -id:A003188 - cp's OEIS frontend

A292203 Primes as they appear in A003188.

3, 2, 7, 5, 13, 11, 31, 29, 23, 19, 17, 53, 61, 59, 41, 43, 47, 37, 97, 103, 101, 109, 107, 127, 113, 83, 89, 73, 79, 71, 67, 193, 199, 197, 223, 211, 241, 251, 233, 239, 229, 227, 163, 167, 173, 191, 181, 179, 151, 149, 157, 137, 139, 131, 389, 397, 409, 401, 433, 439

Offset: 1

Robert G. Wilson v, Sep 11 2017

nonn

a(n) is the m-th prime where m takes the values 2, 1, 4, 3, 6, 5, 11, 10, 9, 8, 7, 18, 15, 16, 17, 12, ... .

Cf. A003188, A292204, A292205.

f[n_] := BitXor[n, Floor[n/2]]; Select[Array[f, 300], PrimeQ]

A307693 Rectangular quotient array, R, of A003188 read by descending antidiagonals; see Comments.

Original entry on oeis.org

1, 3, 1, 2, 3, 1, 6, 2, 2, 1, 7, 6, 4, 3, 1, 5, 7, 5, 2, 3, 1, 4, 5, 3, 6, 2, 2, 1, 12, 4, 8, 7, 5, 4, 2, 1, 13, 12, 9, 5, 6, 5, 4, 3, 1, 15, 13, 10, 4, 4, 3, 3, 2, 3, 1, 14, 15, 7, 12, 10, 8, 7, 6, 2, 3, 1, 10, 14, 6, 13, 11, 9, 9, 7, 6, 2, 2, 1, 11, 10, 16

Offset: 1

Clark Kimberling, Oct 26 2019

Suppose that P = (p(m)) is a permutation of the positive integers, such as A038722. For each n >= 1, let q(n,k) be the k-th index m such that n divides p(m), and let r(n) = p(q(n,k))/n. Let R be the array having (r(n)) as row n. We call R the quotient array of P. Every row of R is a permutation of the positive integers.

In the present case that P = A003188, every row occurs infinitely many times. Specifically, if p is a prime (A000040), then for every multiple m*p of p, the rows numbered m*p are identical. See A327314 for the array that results by deleting duplicate rows from R.

			A003188 = (1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 11, 9, 8, 24, 25, 27, 26, 30, 31, 29, 28, 20, ...)
Row 1 of R is just A003188. To get row 2 of R, skip the odds in A003188 and divide the evens by 2; row 2 equals row 1. Generally, to get row n, divide A003188 by n and then delete the non-integers.
________________
Northwest corner of R:
  1   3   2   6   7   5    4   12   13   15
  1   3   2   6   7   5    4   12   13   15
  1   2   4   5   3   8    9   10    7    6
  1   3   2   6   7   5    4   12   13   15
  1   3   2   5   6   4   10   11   12    8
  1   2   4   5   3   8    9   10    7    6

Cf. A000040, A003188, A327314.

s = Table[BitXor[n, Floor[n/2]], {n, 300}]  (* A003188 *)
g[n_] := Flatten[Position[Mod[s, n], 0]];
u[n_] := s[[g[n]]]/n;
TableForm[Table[Take[u[n], 10], {n, 1, 20}]]  (* A307693 array *)
v[n_, k_] := u[n][[k]]
Table[v[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten (* A307693 sequence *)

A322018 a(n) = A006068(A129760(A003188(n))).

Original entry on oeis.org

0, 0, 3, 0, 7, 4, 7, 0, 15, 8, 11, 8, 15, 12, 15, 0, 31, 16, 19, 16, 23, 20, 23, 16, 31, 24, 27, 24, 31, 28, 31, 0, 63, 32, 35, 32, 39, 36, 39, 32, 47, 40, 43, 40, 47, 44, 47, 32, 63, 48, 51, 48, 55, 52, 55, 48, 63, 56, 59, 56, 63, 60, 63, 0, 127, 64, 67, 64, 71, 68, 71, 64, 79, 72, 75, 72, 79, 76, 79, 64, 95, 80, 83, 80, 87, 84, 87, 80

Offset: 0

Antti Karttunen, Nov 27 2018

nonn

For all n, A207901(a(n)) divides A207901(n), and similarly for A302033.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A003188, A006068, A035327, A129760.

Cf. also A207901, A302033, A322017.

A003188(n) = bitxor(n, n>>1);
A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ After code in A006068
A129760(n) = bitand(n,n-1); \\ From A129760
A322018(n) = A006068(A129760(A003188(n)));

a(n) = A006068(A129760(A003188(n))).

A323895 Dirichlet inverse of binary Gray code, A003188.

Original entry on oeis.org

1, -3, -2, 3, -7, 7, -4, -3, -9, 27, -14, -10, -11, 15, 20, 3, -25, 35, -26, -45, -15, 55, -28, 13, 28, 43, 22, -24, -19, -91, -16, -3, 7, 99, 6, -58, -55, 103, -8, 63, -61, 55, -62, -94, 71, 111, -56, -16, -25, -148, 58, -73, -47, -107, 152, 33, 67, 75, -38, 196, -35, 63, 148, 3, 57, -53, -98, -171, 9, -59, -100, 78, -109, 219, -208, -178, 5, 13

Offset: 1

Antti Karttunen, Feb 08 2019

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

Cf. A003188, A323896.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003188(n) = bitxor(n, n>>1);
v323895 = DirInverseCorrect(vector(up_to,n,A003188(n)));
A323895(n) = v323895[n];

A363710 a(n) is the number of pairs of nonnegative integers (x, y) such that x + y = n and A003188(x) AND A003188(y) = 0 (where AND denotes the bitwise AND operator).

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 2, 2, 4, 8, 6, 4, 6, 4, 2, 2, 4, 8, 10, 8, 12, 12, 6, 4, 6, 12, 8, 4, 6, 4, 2, 2, 4, 8, 10, 8, 16, 20, 10, 8, 12, 24, 20, 12, 16, 12, 6, 4, 6, 12, 16, 12, 16, 16, 8, 4, 6, 12, 8, 4, 6, 4, 2, 2, 4, 8, 10, 8, 16, 20, 10, 8, 16, 32, 28, 20, 28

Offset: 0

Rémy Sigrist, Jun 17 2023

Equivalently, a(n) is the number of k >= 0 such that A332497(k) + A332498(k) = n.

The set of pairs of nonnegative integers (x, y) such that A003188(x) AND A003188(y) = 0 is related to the T-square fractal (see illustration in Links section).

			For n = 8:
- we have:
  k  A332497(8-k)  A332497(k)  A332497(8-k) AND A332497(k)
  -  ------------  ----------  ---------------------------
  0            12           0                            0
  1             4           1                            0
  2             5           3                            1
  3             7           2                            2
  4             6           6                            6
  5             2           7                            2
  6             3           5                            1
  7             1           4                            0
  8             0          12                            0
- so a(8) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of (x, y) such that x, y < 2^10 and A003188(x) AND A003188(y) = 0
Wikipedia, T-square (fractal)

Cf. A003188, A075427, A131218, A332497, A332498.

A131218:= func< n,k | BitwiseAnd(BitwiseXor(n, ShiftRight(n, 1)), BitwiseXor(k, ShiftRight(k, 1))) eq 0 select 1 else 0 >;
A363710:= func< n | (&+[A131218(n-k,k): k in [0..n]]) >;
[A363710(n): n in [0..100]]; // G. C. Greubel, Sep 06 2025

A131218[n_, k_]:= Boole[BitAnd[BitXor[n, BitShiftRight[n, 1]], BitXor[k, BitShiftRight[k, 1]]] ==0];
A363710[n_]:= A363710[n]= Sum[A131218[n-k,k], {k,0,n}];
Table[A363710[n], {n,0,100}] (* G. C. Greubel, Sep 06 2025 *)

a(n) = 2*sum(k=0, n\2, bitand(bitxor(n-k, (n-k)\2), bitxor(k, k\2))==0) - (n==0)

A363710=lambda n: sum(map(lambda k: not (k^k>>1)&(n-k^n-k>>1),range(n+1>>1)))<<1 if n else 1 # Natalia L. Skirrow, Jun 22 2023

def A363710(n): return sum(int( ((n-k)^^((n-k)>>1)) & (k^^(k>>1)) ==0) for k in range(n+1))
print([A363710(n) for n in range(101)]) # G. C. Greubel, Sep 06 2025

a(n) = 2 iff n belongs to A075427.

A099891 XOR difference triangle of A003188 (Gray code numbers), read by rows.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 2, 1, 3, 0, 6, 4, 5, 6, 6, 7, 1, 5, 0, 6, 0, 5, 2, 3, 6, 6, 0, 0, 4, 1, 3, 0, 6, 0, 0, 0, 12, 8, 9, 10, 10, 12, 12, 12, 12, 13, 1, 9, 0, 10, 0, 12, 0, 12, 0, 15, 2, 3, 10, 10, 0, 0, 12, 12, 0, 0, 14, 1, 3, 0, 10, 0, 0, 0, 12, 0, 0, 0, 10, 4, 5, 6, 6, 12, 12, 12, 12, 0, 0, 0, 0, 11, 1

Offset: 0

Paul D. Hanna, Oct 29 2004

Main diagonal is A099892, the XOR BINOMIAL transform of A003188. See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

			Rows begin:
[0],
[1,1],
[3,2,3],
[2,1,3,0],
[6,4,5,6,6],
[7,1,5,0,6,0],
[5,2,3,6,6,0,0],
[4,1,3,0,6,0,0,0],
[12,8,9,10,10,12,12,12,12],
...
where A003188 fills the leftmost column.

Cf. A047999, A003188 (column k=0), A006519 (column k=1), A099892 (diagonal n=k).

Other triangles: A099884, A099889, A099893.

{T(n,k)=local(B);B=0;for(i=0,k,B=bitxor(B,binomial(k,i)%2*(bitxor((n-i),(n-i)\2))));B}

T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*(A003188(n-i)), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i). T(2^n, 2^n) = 3*2^(n-1) for n>0, with T(1, 1)=1 and T(k, k)=0 elsewhere.

T(n,1) = A006519(n), the lowest 1-bit of n (see formula by Franklin T. Adams-Watters in A003188). - Kevin Ryde, Jul 02 2020

A192483 G.f.: A(x) = Sum_{n>=0} x^n * A(x)^A003188(n) where A003188(n) = n XOR floor(n/2).

Original entry on oeis.org

1, 1, 2, 6, 18, 61, 220, 822, 3157, 12378, 49345, 199441, 815467, 3367153, 14020938, 58811032, 248260925, 1053893607, 4496248445, 19268100048, 82902438819, 357987967157, 1550951132419, 6739554074740, 29366902576469, 128287060703669

Offset: 0

Paul D. Hanna, Jul 01 2011

nonn

A003188(n) is the decimal equivalent of the binary Gray code for n; A003188 forms a permutation of the nonnegative integers.

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 18*x^4 + 61*x^5 + 220*x^6 +...
The g.f. A(x) satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^2 + x^4*A(x)^6 + x^5*A(x)^7 + x^6*A(x)^5 + x^7*A(x)^4 + x^8*A(x)^12 + x^9*A(x)^13 + x^10*A(x)^15 +...
where the powers of A(x) are given by A003188, which begins:
[0,1,3,2,6,7,5,4,12,13,15,14,10,11,9,8,24,25,27,26,30,31,29,...].

Cf. A003188, A191769.

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m*(A+x*O(x^n))^bitxor(m,m\2)));polcoeff(A,n)}

A327314 Rectangular array read by descending antidiagonals: the distinct rows of the quotient array, A307693, of A003188.

Original entry on oeis.org

1, 3, 1, 2, 2, 1, 6, 4, 3, 1, 7, 5, 2, 2, 1, 5, 3, 5, 4, 2, 1, 4, 8, 6, 3, 5, 2, 1, 12, 9, 4, 7, 4, 4, 3, 1, 13, 10, 10, 9, 3, 3, 2, 3, 1, 15, 7, 11, 8, 9, 8, 6, 2, 2, 1, 14, 6, 12, 6, 10, 9, 7, 6, 5, 2, 1, 10, 16, 8, 5, 11, 7, 5, 5, 4, 4, 2, 1, 11, 17, 9

Offset: 1

Clark Kimberling, Nov 01 2019

Each row of the quotient array, A307693, occurs infinitely many times. Specifically, if p is a prime (A000040), then for every multiple m*p of p, the rows numbered m*p are identical. In the present array only the first occurrence of each row of A307693 is retained; these are the prime-numbered rows of A307693. Every row is a permutation of the positive integers, so that every positive integer occurs infinitely many times.

			Northwest corner:
  1   3   2   6   7   5   4   12  13  15
  1   2   4   5   3   8   9   10   7   6
  1   3   2   5   6   4  10   11  12   8
  1   2   4   3   7   9   8    6   5  14
  1   2   5   4   3   9  10   11   8   7
  1   2   4   3   8   9   7    6   5  15

Cf. A000040, A003188, A307693.

s = Table[BitXor[n, Floor[n/2]], {n, 2000}]; (* A003188 *)
g[n_] := Flatten[Position[Mod[s, n], 0]];
u[n_] := s[[g[Prime[n]]]]/Prime[n];
Column[Table[Take[u[n], 20], {n, 1, 20}]]  (* A326925 array *)
v[n_, k_] := u[n][[k]];
Table[v[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten  (* A326925 sequence *)

A363930 Irregular table T(n, k), n >= 0, k = 1..A363710(n), read by rows; the n-th row lists the nonnegative numbers m <= n such that A003188(m) AND A003188(n-m) = 0 (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 3, 4, 0, 1, 4, 5, 0, 6, 0, 7, 0, 1, 7, 8, 0, 1, 2, 3, 6, 7, 8, 9, 0, 2, 3, 7, 8, 10, 0, 3, 8, 11, 0, 1, 3, 9, 11, 12, 0, 1, 12, 13, 0, 14, 0, 15, 0, 1, 15, 16, 0, 1, 2, 3, 14, 15, 16, 17, 0, 2, 3, 4, 6, 12, 14, 15, 16, 18, 0, 3, 4, 7, 12, 15, 16, 19

Offset: 0

Rémy Sigrist, Jun 28 2023

This sequence is related to the T-square fractal (see A363710).

			Table T(n, k) begins:
  n   n-th row
  --  ----------------------
   0  0
   1  0, 1
   2  0, 2
   3  0, 3
   4  0, 1, 3, 4
   5  0, 1, 4, 5
   6  0, 6
   7  0, 7
   8  0, 1, 7, 8
   9  0, 1, 2, 3, 6, 7, 8, 9
  10  0, 2, 3, 7, 8, 10
  11  0, 3, 8, 11
  12  0, 1, 3, 9, 11, 12
  13  0, 1, 12, 13
  14  0, 14
  15  0, 15
  16  0, 1, 15, 16

Rémy Sigrist, Table of n, a(n) for n = 0..13126 (rows for n = 0..2^9 flattened)
Wikipedia, T-square (fractal)

See A295989, A353174 and A362327 for similar sequences.

Cf. A003188, A363710.

row(n) = { select (m -> bitand(bitxor(m, m\2), bitxor(n-m, (n-m)\2))==0, [0..n]) }

T(n, 1) = 0.
T(n, A363710(n)) = n.
T(n, k) + T(n, A363710(n)+1-k) = n.

A204686 G.f.: A(x) = Sum_{n>=0} x^n * A(x)^A006068(n) where A006068 forms the inverse permutation of the binary Gray code numbers (A003188).

Original entry on oeis.org

1, 1, 2, 6, 18, 62, 228, 869, 3410, 13663, 55689, 230276, 963851, 4076429, 17394641, 74798044, 323797908, 1409980003, 6171824159, 27141256911, 119854448404, 531261779875, 2362873352568, 10541865261557, 47165485163426, 211572830443060, 951338664998601, 4287174496933723

Offset: 0

Paul D. Hanna, Jan 18 2012

nonn

A006068 satisfies: A006068(n) XOR [A006068(n)/2] = n.

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 18*x^4 + 62*x^5 + 228*x^6 +...
The g.f. A(x) satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^2 + x^4*A(x)^7 + x^5*A(x)^6 + x^6*A(x)^4 + x^7*A(x)^5 + x^8*A(x)^15 + x^9*A(x)^14 + x^10*A(x)^12 +...
where the powers of A(x) are given by A006068, which begins:
[0,1,3,2,7,6,4,5,15,14,12,13,8,9,11,10,31,30,28,29,24,25,27,26,...].

Cf. A192483, A006068, A003188.

{A006068(n)=local(B=n);for(k=1,floor(log(n+1)/log(2)),B=bitxor(B,n\2^k));B}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*(A+x*O(x^n))^A006068(m))); polcoeff(A, n)}

cp's OEIS Frontend

A292203 Primes as they appear in A003188.

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Mathematica

A307693 Rectangular quotient array, R, of A003188 read by descending antidiagonals; see Comments.

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Mathematica

A322018 a(n) = A006068(A129760(A003188(n))).

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A323895 Dirichlet inverse of binary Gray code, A003188.

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A363710 a(n) is the number of pairs of nonnegative integers (x, y) such that x + y = n and A003188(x) AND A003188(y) = 0 (where AND denotes the bitwise AND operator).

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A099891 XOR difference triangle of A003188 (Gray code numbers), read by rows.

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A192483 G.f.: A(x) = Sum_{n>=0} x^n * A(x)^A003188(n) where A003188(n) = n XOR floor(n/2).

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A327314 Rectangular array read by descending antidiagonals: the distinct rows of the quotient array, A307693, of A003188.

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A363930 Irregular table T(n, k), n >= 0, k = 1..A363710(n), read by rows; the n-th row lists the nonnegative numbers m <= n such that A003188(m) AND A003188(n-m) = 0 (where AND denotes the bitwise AND operator).