A186541 -id:A186541 - cp's OEIS frontend

A186542 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186541.

1, 5, 7, 10, 13, 16, 19, 21, 24, 27, 29, 32, 35, 38, 40, 43, 46, 49, 51, 54, 57, 60, 62, 65, 68, 71, 73, 76, 79, 81, 84, 87, 90, 92, 95, 98, 101, 103, 106, 109, 111, 114, 117, 120, 122, 125, 128, 131, 133, 136, 139, 142, 144, 147, 150, 152, 155, 158, 161, 163, 166, 169, 172, 174, 177, 180, 183, 185, 188, 191, 193, 196, 199, 202, 204, 207, 210, 213, 215, 218, 221, 224, 226, 229, 232, 234, 237, 240

Offset: 1

Clark Kimberling, Feb 23 2011

nonn

See A186219 for a discussion of adjusted joint rank sequences.

			First, write
1..4..9..16..25..36..49..... (i^2)
.........10.....25.....46.. (-2+3j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 after -2+3j^2:
a=(2,3,4,6,8,9,11,12,14,15,17,18,..)=A186541
b=(1,5,7,10,13,16,19,21,24,27,29...)=A186542.

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A186219, A186539, A186540, A186541.

Mathematica
```
(See A186541.)
```

a(n)=n+floor(sqrt((1/3)n^2+5/6))=A186541(n).

b(n)=n+floor(sqrt(3n^2-5/2))=A186542(n).

A186539 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186540.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 48, 50, 52, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 82, 83, 85, 86, 88, 89, 91, 93, 94, 96, 97, 99, 100, 102, 104, 105, 107, 108, 110, 111, 113, 115, 116, 118, 119, 121, 123, 124, 126, 127, 129, 130, 132, 134, 135, 137, 138, 140, 141, 143, 145, 146, 148, 149, 151, 153, 154, 156, 157

Offset: 1

Clark Kimberling, Feb 23 2011

nonn

See A186219 for a discussion of adjusted joint rank sequences.

Differs from A059555 at n=97, 123, 194, 220, 246, ... - R. J. Mathar, May 18 2020

			First, write
1..4..9..16..25..36..49.... (i^2)
.......10....25.....46.. (-2+3j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 before -2+3j^2:
a=(1,3,4,6,7,9,11,12,14,15,17,18,..)=A186539
b=(2,5,8,10,13,16,19,21,24,27,30...)=A186540.

Cf. A186219, A186540, A186541, A186542.

(* adjusted joint rank sequences a and b, using general formula for ranking ui^2+vi+w and xj^2+yj+z *)
d = 1/2; u = 1; v = 0; w = 0; x = 3; y = 0; z = -2;
h[n_] := -y + (4 x (u*n^2 + v*n + w - z - d) + y^2)^(1/2);
a[n_] := n + Floor[h[n]/(2 x)];
k[n_] := -v + (4 u (x*n^2 + y*n + z - w + d) + v^2)^(1/2);
b[n_] := n + Floor[k[n]/(2 u)];
Table[a[n], {n, 1, 100}]  (* A186539 *)
Table[b[n], {n, 1, 100}]  (* A186540 *)

a(n)=n+floor(sqrt((1/3)n^2+1/24))=A186539(n).

b(n)=n+floor(sqrt(3n^2-3/2))=A186540(n).

A186540 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186539.

Original entry on oeis.org

2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 30, 32, 35, 38, 40, 43, 46, 49, 51, 54, 57, 60, 62, 65, 68, 71, 73, 76, 79, 81, 84, 87, 90, 92, 95, 98, 101, 103, 106, 109, 112, 114, 117, 120, 122, 125, 128, 131, 133, 136, 139, 142, 144, 147, 150, 152, 155, 158, 161, 163, 166, 169, 172, 174, 177, 180, 183, 185, 188, 191, 193, 196, 199, 202, 204, 207, 210, 213, 215, 218

Offset: 1

Clark Kimberling, Feb 23 2011

nonn

See A186219 for a discussion of adjusted joint rank sequences.

Does this differ from A054088? The first 42000 entries of both sequences at least are the same. - R. J. Mathar, Feb 25 2011

			First, write
1..4..9..16..25..36..49.. (i^2)
.......10....25....46.. (-2+3j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 before -2+3j^2:
b=(1,3,4,6,7,9,11,12,14,15,17,18,..)=A186539
a=(2,5,8,10,13,16,19,21,24,27,30...).

Cf. A186219, A186539, A186541, A186542.

Mathematica
```
(* See A186539 *)
```

b(n) = n+floor(sqrt((1/3)n^2+1/24)) = A186539(n).

a(n) = n+floor(sqrt(3n^2-3/2)).

cp's OEIS Frontend

A186542 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186541.

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A186539 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186540.

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A186540 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186539.

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