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 1-5 of 5 results.

A186274 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the triangular numbers and octagonal numbers. Complement of A186159.

Original entry on oeis.org

2, 5, 9, 12, 15, 19, 22, 26, 29, 33, 36, 40, 43, 46, 50, 53, 57, 60, 64, 67, 71, 74, 78, 81, 84, 88, 91, 95, 98, 102, 105, 109, 112, 115, 119, 122, 126, 129, 133, 136, 140, 143, 147, 150, 153, 157, 160, 164, 167, 171, 174, 178, 181, 184, 188, 191, 195, 198, 202, 205, 209, 212, 216, 219, 222, 226, 229, 233, 236, 240, 243, 247, 250, 253, 257, 260, 264, 267, 271, 274, 278, 281, 284, 288, 291, 295, 298, 302, 305, 309, 312, 316, 319, 322, 326, 329, 333, 336, 340, 343
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2011

Keywords

Comments

See A186159.

Examples

			First, write the triangular and octagonal numbers:
1..3..6.....10..15..21..28
1........8..........21......
Then replace each by its rank, where ties are settled by ranking the triangular number before the octagonal:
a=(1,3,4,6,7,8,10,11,13,...)=A186159.
b=(2,5,9,12,15,19,22,26,...)=A186274.

Crossrefs

Cf. A186159, A186219.

Programs

A186145 Rank of n^2 when {i^2: i>=1} and {j^3: j>=1} are jointly ranked with i^2 before j^3 when i^2=j^3. Complement of A186146.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 106, 107, 108, 110, 111, 112, 113, 114, 115, 116, 118, 119, 120, 121
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2011

Keywords

Comments

Suppose u,v,p,q are positive integers and 0<|d|<1. Let
a(n)=n+floor(((u*n^p-d)/v)^(1/q)),
b(n)=n+floor(((v*n^q+d)/u)^(1/p)).
When the disjoint sets {u*i^p} and {v*j^q+d} are jointly ranked, the rank of u*n^p is a(n) and the rank of v*n^q+d is b(n). Therefore a and b are a pair of complementary sequences. Choosing d carefully serves as a basis for two types of adjusted joint rankings of non-disjoint sets {u*i^p} and {v*j^q}.
First, if we place u*i^p before v*j^q whenever u*i^p=v*j^q, then with 0
More generally, if u=h/k and v=s/t are positive rational numbers in lowest terms, then a(n) and b(n) are the respective ranks of u*n^p and v*n^q, adjusted as described above, according as d=1/(2kq) or d=-1/(2kq). Examples: A186148-A186159.

Examples

			Write the squares and cubes thus:
1..4....9..16..25....36..49..64..81
1.....8...........27.........64.....
Replace each by its rank, where ties are settled by ranking the square before the cube:
a=(1,3,5,6,7,9,10,11,13,...)
b=(2,4,8,12,...)

Crossrefs

Cf. A186146.

Programs

  • Mathematica
    d=1/2;
a[n_]:=n+Floor[(n^2-d)^(1/3)]; (* rank of n^2 *)
b[n_]:=n+Floor[(n^3+d)^(1/2)]; (* rank of n^3+1/2 *)
Table[a[n],{n,1,100}]
Table[b[n],{n,1,100}]
(* end *)
(* A more general program follows. *)
d=1/2; u=1; v=1; p=2; q=3;
h[n_]:=((u*n^p-d)/v)^(1/q);
a[n_]:=n+Floor[h[n]]; (* rank of u*n^p *)
k[n_]:=((v*n^q+d)/u)^(1/p);
b[n_]:=n+Floor[k[n]]; (* rank of v*n^q *)
Table[a[n],{n,1,100}]
Table[b[n],{n,1,100}]

Formula

a(n)=n+floor((n^2-1/2)^(1/3)) (A186145).
b(n)=n+floor((n^3+1/2)^(1/2)) (A186146).

A186275 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the triangular numbers and octagonal numbers. Complement of A186276.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 10, 11, 13, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 77, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 94, 96, 97, 99, 100, 101, 103, 104, 106, 107, 108, 110, 111, 113, 114, 116, 117, 118, 120, 121, 123, 124, 125, 127, 128, 130, 131, 132, 134, 135, 137, 138, 139, 141
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2011

Keywords

Comments

See A186159.

Examples

			First, write the triangular and octagonal numbers:
1..3..6.....10..15..21..28
1........8..........21......
Then replace each by its rank, where ties are settled by ranking the triangular number after the octagonal:
a=(2,3,4,6,7,9,10,11,13,...)=A186275.
b=(1,5,8,12,15,19,22,26,...)=A186276.

Crossrefs

Cf. A186159, A186274, A186276.

Programs

  • Mathematica
    (* adjusted joint ranking; general formula *)
d=-1/2; u=1/2; v=1/2; w=0; x=3; y=-2; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n], {n, 1, 100}] (* A186275 *)
Table[b[n], {n, 1, 100}] (* A186276 *)

A186276 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the triangular numbers and octagonal numbers. Complement of A186275.

Original entry on oeis.org

1, 5, 8, 12, 15, 19, 22, 26, 29, 33, 36, 40, 43, 46, 50, 53, 57, 60, 64, 67, 71, 74, 78, 81, 84, 88, 91, 95, 98, 102, 105, 109, 112, 115, 119, 122, 126, 129, 133, 136, 140, 143, 147, 150, 153, 157, 160, 164, 167, 171, 174, 178, 181, 184, 188, 191, 195, 198, 202, 205, 209, 212, 215, 219, 222, 226, 229, 233, 236, 240, 243, 247, 250, 253, 257, 260, 264, 267, 271, 274, 278, 281, 284, 288, 291, 295, 298, 302, 305, 309, 312, 316, 319, 322, 326, 329, 333, 336, 340, 343
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2011

Keywords

Comments

See A186275.

Examples

			First, write the triangular and octagonal numbers:
1..3..6.....10..15..21..28
1........8..........21......
Then replace each by its rank, where ties are settled by ranking the triangular number after the octagonal:
a=(2,3,4,6,7,9,10,11,13,...)=A186275.
b=(1,5,8,12,15,19,22,26,...)=A186276.

Crossrefs

Cf. A186159, A186274, A186275.

Programs

A186156 Rank of n^3 when {i^3: i>=1} and {2j^2: j>=1} are jointly ranked with i^3 before 2j^2 when i^3=2j^2. Complement of A186157.

Original entry on oeis.org

1, 3, 6, 9, 12, 16, 20, 23, 28, 32, 36, 41, 46, 51, 56, 61, 66, 71, 77, 83, 89, 94, 100, 107, 113, 119, 126, 132, 139, 146, 153, 159, 167, 174, 181, 188, 196, 203, 211, 218, 226, 234, 242, 250, 258, 266, 274, 283, 291, 299, 308, 317, 325, 334, 343, 352, 361, 370, 379, 388, 397, 407, 416, 426, 435, 445, 454, 464, 474, 484, 494, 503, 514, 524, 534, 544, 554, 565, 575, 585, 596, 607, 617, 628, 639, 649, 660, 671, 682
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2011

Keywords

Comments

See A186145 for a discussion of adjusted joint rank sequences.

Examples

			Write separate rankings as
1....8.....27........64........125...
..2..8..18....32..50....72..98.....128...
Then replace each number by its rank, where ties are settled by ranking i^3 before 2j^2.

Crossrefs

Cf. A186145, A186157, A186158, A186159.

Programs

  • Mathematica
    d=1/2; u=1; v=2; p=3; q=2;
h[n_]:=((u*n^p-d)/v)^(1/q);
a[n_]:=n+Floor[h[n]]; (* rank of u*n^p *)
k[n_]:=((v*n^q+d)/u)^(1/p);
b[n_]:=n+Floor[k[n]]; (* rank of v*n^q *)
Table[a[n],{n,1,100}] (* A186156 *)
Table[b[n],{n,1,100}] (* A186157 *)

Formula

a(n)=n+floor(((n^3-1/2)/2)^(1/2)), A186156.
b(n)=n+floor((2n^2+1/2)^(1/3)), A186157.
Showing 1-5 of 5 results.

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