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.

A356181 a(n) = A054406(A001951(n)).

Original entry on oeis.org

2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, 49, 52, 56, 59, 61, 66, 68, 73, 75, 78, 82, 85, 89, 92, 97, 99, 101, 106, 108, 113, 115, 118, 123, 125, 130, 132, 134, 139, 141, 146, 149, 153, 156, 158, 163, 165, 170, 172, 175, 179, 182, 186, 189, 194
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2022

Keywords

Comments

This is the second of four sequences that partition the positive integers. See A356180.

Examples

			(1)  v o u = (1, 3, 6, 8, 12, 13, 15, 19, 20, 24, 25, 27, 31, 32, ...) = A356180
(2)  v' o u = (2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, ...) = A356181
(3)  v o u' = (5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, ...) = A356182
(4)  v' o u' = (7, 14, 23, 30, 40, 47, 54, 63, 70, 80, 87, 94, 104, ...) = A356183
		

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308 (intersections), A356088 (reverse composites), A356180, A356182, A356183.

Programs

  • Mathematica
    z = 800; zz = 100;
    u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
    u1 = Complement[Range[Max[u]], u];       (* A001952 *)
    v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
    v1 = Complement[Range[Max[v]], v];  (* A054406 *)
    Table[v[[u[[n]]]], {n, 1, zz}]      (* A356180 *)
    Table[v1[[u[[n]]]], {n, 1, zz}]     (* A356181 *)
    Table[v[[u1[[n]]]], {n, 1, zz}]     (* A356182 *)
    Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A356183 *)

A356182 a(n) = A022838(A001952(n)).

Original entry on oeis.org

5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, 81, 88, 93, 100, 105, 110, 117, 122, 129, 135, 140, 147, 152, 159, 164, 171, 176, 181, 188, 193, 200, 206, 211, 218, 223, 230, 235, 240, 247, 252, 259, 265, 271, 277, 282, 289, 294, 301, 306, 311, 318, 323
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2022

Keywords

Comments

This is the third of four sequences that partition the positive integers. See A356180.

Examples

			(1)  v o u = (1, 3, 6, 8, 12, 13, 15, 19, 20, 24, 25, 27, 31, 32, ...) = A356180
(2)  v' o u = (2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, ...) = A356181
(3)  v o u' = (5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, ...) = A356182
(4)  v' o u' = (7, 14, 23, 30, 40, 47, 54, 63, 70, 80, 87, 94, 104, ...) = A356183
		

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308 (intersections), A356088 (reverse composites), A356180, A356181, A356183.

Programs

  • Mathematica
    z = 800; zz = 100;
    u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
    u1 = Complement[Range[Max[u]], u];       (* A001952 *)
    v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
    v1 = Complement[Range[Max[v]], v];  (* A054406 *)
    Table[v[[u[[n]]]], {n, 1, zz}]      (* A356180 *)
    Table[v1[[u[[n]]]], {n, 1, zz}]     (* A356181 *)
    Table[v[[u1[[n]]]], {n, 1, zz}]     (* A356182 *)
    Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A356183 *)
  • Python
    from math import isqrt
    def A356182(n): return isqrt(3*((k:=n<<1)+isqrt(k*n))**2) # Chai Wah Wu, Sep 05 2022

A356183 a(n) = A054406(A001952(n)).

Original entry on oeis.org

7, 14, 23, 30, 40, 47, 54, 63, 70, 80, 87, 94, 104, 111, 120, 127, 137, 144, 151, 160, 167, 177, 184, 191, 201, 208, 217, 224, 234, 241, 248, 257, 264, 274, 281, 288, 298, 305, 314, 321, 328, 338, 345, 354, 362, 371, 378, 385, 395, 402, 411, 418, 425, 435
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2022

Keywords

Comments

This is the fourth of four sequences that partition the positive integers. See A356180.

Examples

			(1)  v o u = (1, 3, 6, 8, 12, 13, 15, 19, 20, 24, 25, 27, 31, 32, ...) = A356180
(2)  v' o u = (2, 4, 9, 11, 16, 18, 21, 26, 28, 33, 35, 37, 42, 44, ...) = A356181
(3)  v o u' = (5, 10, 17, 22, 29, 34, 39, 46, 51, 58, 64, 69, 76, ...) = A356182
(4)  v' o u' = (7, 14, 23, 30, 40, 47, 54, 63, 70, 80, 87, 94, 104, ...) = A356183
		

Crossrefs

Cf. A001951, A001952, A022838, A054406, A346308 (intersections), A356088 (reverse composites), A356180, A356181, A356182.

Programs

  • Mathematica
    z = 800; zz = 100;
    u = Table[Floor[n*Sqrt[2]], {n, 1, z}];  (* A001951 *)
    u1 = Complement[Range[Max[u]], u];       (* A001952 *)
    v = Table[Floor[n*Sqrt[3]], {n, 1, z}];  (* A022838 *)
    v1 = Complement[Range[Max[v]], v];  (* A054406 *)
    Table[v[[u[[n]]]], {n, 1, zz}]      (* A356180 *)
    Table[v1[[u[[n]]]], {n, 1, zz}]     (* A356181 *)
    Table[v[[u1[[n]]]], {n, 1, zz}]     (* A356182 *)
    Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A356183 *)
Showing 1-3 of 3 results.