A186145 -id:A186145 - cp's OEIS frontend

A186219 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 squares. Complement of A186220.

1, 3, 5, 7, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 58, 60, 61, 63, 65, 66, 68, 70, 72, 73, 75, 77, 78, 80, 82, 83, 85, 87, 89, 90, 92, 94, 95, 97, 99, 101, 102, 104, 106, 107, 109, 111, 113, 114, 116, 118, 119, 121, 123, 124, 126, 128, 130, 131, 133, 135, 136, 138, 140, 142, 143, 145, 147, 148, 150, 152, 153, 155, 157, 159, 160, 162, 164, 165, 167, 169, 171

Offset: 1

Clark Kimberling, Feb 15 2011

nonn

Suppose that f and g are strictly increasing functions for which (f(i)) and (g(j)) are integer sequences.  If 0<|d|<1, the sets F={f(i): i>=1} and G={g(j)+d: j>=1} are clearly disjoint.  Let f^=(inverse of f) and g^=(inverse of g).  When the numbers in F and G are jointly ranked, the rank of f(n) is a(n):=n+floor(g^(f(n))-d), and the rank of g(n)+d is b(n):=n+floor(f^(g(n))+d).  Therefore, the sequences a and b are a complementary pair.
Although the sequences (f(i)) and (g(j)) may not be disjoint, the sequences (f(i)) and (g(j)+d) are disjoint, and this observation enables two types of adjusted joint rankings:
(1) if 0Using f(i)=ui^2+vi+w and g(j)=xj^2+yj+z, we can carry out adjusted joint rankings of any pair of polygonal sequences (triangular, square, pentagonal, etc.)  In this case,
  a(n)=n+floor((-y+sqrt(4x(un^2+vn+w-z-d)+y^2))/(2x)),
  b(n)=n+floor((-v+sqrt(4u(xn^2+yn+z-w+d)+v^2)/(2u)),
  where a(n) is the rank of un^2+vn+w and b(n) is the rank
  of xn^2+yn+z+d, where d must be chosen small enough, in
  absolute value, that the sets F and G are disjoint.
Example:  f=A000217 (triangular numbers) and g=A000290 (squares) yield adjusted rank sequences a=A186219 and b=A186220 for d=1/4 and a=A186221 and b=A186222 for d=-1/4.

			First, write
1..3...6..10..15...21..28..36..45...  (triangular)
1....4.. 9......16...25....36....49.. (square)
Replace each number by its rank, where ties are settled by ranking the triangular number before the square:
a=(1,3,5,7,8,10,12,13,...)
b=(2,4,6,9,11,14,16,18,...).

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A186145 (joint ranks of squares and cubes),
A000217 (triangular numbers),
A000290 (squares),
A186220 (complement of A186119)
A186221 ("after" instead of "before")
A186222 (complement of A186221).

[n + Floor(Sqrt((n^2 + n)/2 - 1/4)): n in [1..100]]; // G. C. Greubel, Aug 26 2018

(* adjusted joint ranking of triangular numbers and squares, using general formula *)
d=1/4; u=1/2; v=1/2; w=0; x=1; y=0; 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)]; (* rank of triangular n(n+1)/2 *)
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)]; (* rank of square n^2 *)
Table[a[n],{n,1,100}] (* A186219 *)
Table[b[n],{n,1,100}] (* A186220 *)

vector(100, n, n + floor(sqrt((n^2 + n)/2 - 1/4))) \\ G. C. Greubel, Aug 26 2018

a(n) = n + floor(sqrt((n^2+n)/2 - 1/4)), (A186219).

b(n) = n + floor((-1 + sqrt(8*n^2+3))/2), (A186220).

A186350 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 odd numbers and the triangular numbers. Complement of A186351.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 138, 139, 140, 141

Offset: 1

Clark Kimberling, Feb 18 2011

nonn

Suppose that f and g are strictly increasing functions for which (f(i)) and (g(j)) are integer sequences.  If 0<|d|<1, the sets F={f(i): i>=1} and G={g(j)+d: j>=1} are clearly disjoint.  Let f^=(inverse of f) and g^=(inverse of g).  When the numbers in F and G are jointly ranked, the rank of f(n) is a(n):=n+floor(g^(f(n))-d), and the rank of g(n)+d is b(n):=n+floor(f^(g(n))+d).  Therefore, the sequences a and b are a complementary pair.
Although the sequences (f(i)) and (g(j)) may not be disjoint, the sequences (f(i)) and (g(j)+d) are disjoint, and this observation enables two types of adjusted joint rankings:
(1) if 0Using f(i)=ui+v, g(j)=xj^2+yj+z, we find a and b given by
  a(n)=n+floor((-y+sqrt(4x(un+v-d)+y^2))/(2x)),
  b(n)=n+floor((xn^2+yn-v+d)/(2u))),
  where a(n) is the rank of un+v and b(n) is the rank
  xn^2+yn+z+d, and d must be chosen small enough, in
  absolute value, that the sets F and G are disjoint.
Example:  f=A000217 (odd numbers) and g=A000290 (triangular numbers) yield adjusted joint rank sequences a=A186350 and b=A186351 for d=1/2 and a=A186352 and b=A186353 for d=-1/2.
For other classes of adjusted joint rank sequences, see A186145 and A186219.

			First, write
1..3..5..7..9..11..13..15..17..21..23.. (odds)
1..3....6.....10.......15......21.... (triangular)
Then replace each number by its rank, where ties are settled by ranking the odd number before the triangjular:
a=(1,3,5,7,8,10,11,12,14,....)=A186350
b=(2,4,6,9,13,17,21,26,32,...)=A186351.

Cf. A186145, A186219, A186351, A186352, A186353,

A005408 (odd numbers), A000217 (triangular numbers).

(* adjusted joint rank sequences a and b, using general formula for ranking 1st degree u*n+v and 2nd degree x*n^2+y*n+z *)
d=1/2; u=2; v=-1; x=1/2; y=1/2; (* odds and triangular *)
h[n_]:=(-y+(4x(u*n+v-d)+y^2)^(1/2))/(2x);
a[n_]:=n+Floor[h[n]]; (* rank of u*n+v *)
k[n_]:=(x*n^2+y*n-v+d)/u;
b[n_]:=n+Floor[k[n]]; (* rank of x*n^2+y*n+d *)
Table[a[n],{n,1,120}]  (* A186350 *)
Table[b[n],{n,1,100}]  (* A186351 *)

a(n)=n+floor(-1/2+sqrt(4n-9/4))=A186350(n).

b(n)=n+floor((n^2+n+3)/4)=A186351(n).

A186152 Rank of (1/8)n^3 when {(1/8)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/8)i^3 before j^2 when (1/8)i^3=j^2. Complement of A186153.

Original entry on oeis.org

1, 2, 4, 6, 8, 11, 13, 15, 18, 21, 23, 26, 29, 32, 35, 38, 41, 44, 48, 51, 55, 58, 61, 65, 69, 72, 76, 80, 84, 88, 92, 95, 100, 104, 108, 112, 116, 120, 125, 129, 133, 138, 142, 147, 151, 156, 160, 165, 170, 174, 179, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234, 239, 245, 250, 255, 260, 266, 271, 277, 282, 287, 293, 299, 304, 310, 315, 321, 327, 332, 338, 344, 350, 356, 362, 367, 373, 379, 385, 391, 397, 403, 410, 416, 422, 428, 434, 440, 447, 453

Offset: 1

Clark Kimberling, Feb 13 2011

nonn

See A186145 for a discussion of adjusted joint rank sequences.

Cf. A186145, A186153.

d=1/16;u=1/8;v=1;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}] (* A186152 *)
Table[b[n],{n,1,100}]  (* A186153 *)

a(n)=n+floor(((1/8)n^3-1/16)^(1/2)), A186152.

b(n)=n+floor(((n^2)/16+1/2)^(1/3)), A186153.

A186148 Rank of (1/4)n^3 when {(1/4)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/4)i^3 before j^2 when (1/4)i^3=j^2. Complement of A186149.

Original entry on oeis.org

1, 3, 5, 7, 10, 13, 16, 19, 22, 25, 29, 32, 36, 40, 44, 47, 52, 56, 60, 64, 69, 73, 78, 82, 87, 92, 97, 102, 107, 112, 117, 122, 127, 133, 138, 143, 149, 155, 160, 166, 172, 178, 183, 189, 195, 201, 208, 214, 220, 226, 233, 239, 245, 252, 258, 265, 272, 278, 285, 292, 299, 306, 313, 319, 327, 334, 341, 348, 355, 362, 370, 377, 384, 392, 399, 407, 414, 422, 430, 437, 445, 453, 461, 468, 476, 484, 492, 500

Offset: 1

Clark Kimberling, Feb 13 2011

See A187645.

			Write preliminary separate rankings:
1/4...2....27/4....16.....125/4...
....1...4.......9..16..25........36..49
Then replace each number by its rank, where ties are settled by ranking the top number before the bottom.

Cf. A186145, A186149.

d=1/8; u=1/4; v=1; 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}] (* A186148 *)
Table[b[n],{n,1,100}] (* A186149 *)

a(n) = n + floor(((1/4)*n^3 - 1/8)^(1/2)).

A186153 Rank of n^2 when {(1/8)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/8)i^3 before j^2 when (1/8)i^3=j^2. Complement of A186152.

Original entry on oeis.org

3, 5, 7, 9, 10, 12, 14, 16, 17, 19, 20, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 96, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 122, 123, 124, 126, 127, 128, 130, 131, 132, 134, 135, 136, 137, 139, 140

Offset: 1

Clark Kimberling, Feb 13 2011

nonn

See A186152.

Cf. A186145, A186152, A186153, A186154.

Mathematica
```
(See A186152.)
```

See A186152.

A186154 Rank of (1/8)n^3 when {(1/8)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/8)i^3 after j^2 when (1/8)i^3=j^2. Complement of A186155.

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 13, 16, 18, 21, 23, 26, 29, 32, 35, 38, 41, 45, 48, 51, 55, 58, 61, 65, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 129, 133, 138, 142, 147, 151, 156, 160, 165, 170, 175, 179, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234, 239, 245, 250, 255, 260, 266, 271, 277, 282, 288, 293, 299, 304, 310, 315, 321, 327, 332, 338, 344, 350, 356, 362, 367, 373, 379, 385, 391, 397, 403, 410

Offset: 1

Clark Kimberling, Feb 13 2011

nonn

A186154 results from changing "before" to "after" in the name of A186152. See A186145 for a discussion of adjusted joint rank sequences.

Cf. A186145, A186155.

d=-1/16; u=1/8; v=1; 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}]  (* A186154 *)
Table[b[n],{n,1,100}]  (* A186155 *)

A186150 Rank of (1/4)n^3 when {(1/4)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/4)i^3 after j^2 when (1/4)i^3=j^2. Complement of A186151.

Original entry on oeis.org

1, 3, 5, 8, 10, 13, 16, 19, 22, 25, 29, 32, 36, 40, 44, 48, 52, 56, 60, 64, 69, 73, 78, 82, 87, 92, 97, 102, 107, 112, 117, 122, 127, 133, 138, 144, 149, 155, 160, 166, 172, 178, 183, 189, 195, 201, 208, 214, 220, 226, 233, 239, 245, 252, 258, 265, 272, 278, 285, 292, 299, 306, 313, 320, 327, 334, 341, 348, 355, 362, 370, 377, 384, 392, 399, 407, 414, 422, 430, 437, 445, 453, 461, 468, 476, 484, 492, 500, 508, 516, 525, 533, 541, 549, 557, 566, 574, 583, 591, 600

Offset: 1

Clark Kimberling, Feb 13 2011

nonn

See A186145.

Cf A186145, A186151.

d=-1/8; u=1/4; v=1; 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}]  (* A186150 *)
Table[b[n],{n,1,100}]  (* A186151 *)

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

Clark Kimberling, Feb 13 2011

nonn

See A186145 for a discussion of adjusted joint rank sequences.

			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.

Cf. A186145, A186157, A186158, A186159.

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 *)

a(n)=n+floor(((n^3-1/2)/2)^(1/2)), A186156.

b(n)=n+floor((2n^2+1/2)^(1/3)), A186157.

A186147 Rank of n^3 when {i^2: i>=1} and {j^3: j>=1} are jointly ranked with i^2 after j^3 when i^2=j^3. Complement of A135674.

Original entry on oeis.org

1, 4, 8, 11, 16, 20, 25, 30, 35, 41, 47, 53, 59, 66, 73, 79, 87, 94, 101, 109, 117, 125, 133, 141, 149, 158, 167, 176, 185, 194, 203, 213, 222, 232, 242, 251, 262, 272, 282, 292, 303, 314, 324, 335, 346, 357, 369, 380, 391, 403, 415, 426, 438, 450, 462, 475, 487, 499, 512, 524, 537, 550, 563, 575, 589, 602, 615, 628, 642, 655, 669, 682, 696, 710, 724, 738, 752, 766, 781, 795, 809, 824, 839, 853, 868, 883, 898, 913

Offset: 1

Clark Kimberling, Feb 13 2011

nonn

See A186145.

Cf. A135674, A186145.

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}]  (* A135674 *)
Table[b[n],{n,1,100}]  (* A186147 *)

b(n):=n+floor((n^3-1/2)^(1/2)), as in the Mathematica program.

Showing 1-10 of 13 results. Next

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A186146 Rank of n^3 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 A186145.

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A186219 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 squares. Complement of A186220.

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A186350 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 odd numbers and the triangular numbers. Complement of A186351.

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A186152 Rank of (1/8)n^3 when {(1/8)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/8)i^3 before j^2 when (1/8)i^3=j^2. Complement of A186153.

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A186148 Rank of (1/4)n^3 when {(1/4)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/4)i^3 before j^2 when (1/4)i^3=j^2. Complement of A186149.

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A186153 Rank of n^2 when {(1/8)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/8)i^3 before j^2 when (1/8)i^3=j^2. Complement of A186152.

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A186154 Rank of (1/8)n^3 when {(1/8)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/8)i^3 after j^2 when (1/8)i^3=j^2. Complement of A186155.

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A186150 Rank of (1/4)n^3 when {(1/4)i^3: i>=1} and {j^2>: j>=1} are jointly ranked with (1/4)i^3 after j^2 when (1/4)i^3=j^2. Complement of A186151.

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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.

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