cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A359352 a(n) = A026430(1 + A026430(n)).

Original entry on oeis.org

3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, 42, 44, 46, 48, 51, 54, 55, 57, 60, 63, 65, 68, 69, 70, 73, 77, 78, 80, 82, 84, 87, 90, 91, 93, 96, 99, 100, 103, 105, 107, 109, 111, 114, 117, 118, 121, 123, 125, 128, 130, 132, 134, 136, 138
Offset: 1

Views

Author

Clark Kimberling, Jan 26 2023

Keywords

Comments

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n) = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.

Examples

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135

Crossrefs

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359353-A360139.

Programs

  • Mathematica
    z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v]; (* A285953 *)
Table[u[[v[[n]]]], {n, 1, zz}]     (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]    (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]    (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]   (* A360135 *)
  • Python
    def A359352(n): return (m:=n+1+(n-1>>1)+(n-1&1|(n.bit_count()&1^1)))+(m-1>>1)+(m-1&1|(m.bit_count()&1^1)) # Chai Wah Wu, Mar 01 2023

A360135 a(n) = A356133(A285953(n+1)).

Original entry on oeis.org

2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, 131, 142, 148, 161, 169, 176, 187, 193, 202, 215, 223, 229, 238, 251, 257, 269, 278, 283, 292, 305, 313, 320, 331, 337, 346, 359, 367, 373, 382, 394, 400, 412, 421, 428, 439, 445, 454, 466, 472
Offset: 1

Views

Author

Clark Kimberling, Jan 30 2023

Keywords

Comments

This is the fourth of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n) = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.

Examples

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135

Crossrefs

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359352-A360134, A360136-A360139.

Programs

  • Mathematica
    z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[u[[v[[n]]]], {n, 1, zz}]      (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]     (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]     (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]    (* A360135 *)

A360134 a(n) = A356133(1 + A026430(n)).

Original entry on oeis.org

4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, 79, 83, 85, 92, 94, 101, 106, 110, 112, 119, 124, 127, 133, 137, 140, 146, 151, 155, 157, 164, 166, 173, 178, 182, 184, 191, 197, 200, 206, 208, 211, 218, 220, 227, 233, 236, 242, 244, 247, 253
Offset: 1

Views

Author

Clark Kimberling, Jan 30 2023

Keywords

Comments

This is the third of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n) = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.

Examples

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135

Crossrefs

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359352, A359353, A360135-A360139.

Programs

  • Mathematica
    z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[u[[v[[n]]]], {n, 1, zz}]      (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]     (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]     (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]    (* A360135 *)
  • Python
    def A360134(n): return 3*(m:=n+1+(n-1>>1)+(n-1&1|(n.bit_count()&1^1)))-(2 if (m-1).bit_count()&1 else 1) # Chai Wah Wu, Mar 01 2023
Showing 1-3 of 3 results.

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