A186148 -id:A186148 - cp's OEIS frontend

A186149 Rank of n^2 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 A186148.

2, 4, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 88, 89, 90, 91, 93, 94, 95, 96, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 110, 111, 113, 114, 115, 116, 118, 119, 120, 121, 123, 124, 125, 126, 128, 129, 130, 131, 132, 134

Offset: 1

Clark Kimberling, Feb 13 2011

nonn

See A186148 and A186145.

			See A186148.

Cf. A186145, A186148.

Mathematica
```
(See A186148.)
```

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

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

Clark Kimberling, Feb 13 2011

nonn

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 0More 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.

			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,...)

Cf. A186146.

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}]

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

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

Showing 1-2 of 2 results.

cp's OEIS Frontend

A186149 Rank of n^2 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 A186148.

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