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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.
14
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
1,2
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<d<1, a(n) and b(n) are the ranks of u*n^p and v*j^q, respectively. For the second type, if we place u*i^p after v*j^q whenever u*i^p=v*j^q, then with -1<d<0, a(n) and b(n) are ranks of u*n^p and v*j^q, respectively.
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.
FORMULA
a(n)=n+floor((n^2-1/2)^(1/3)) (A186145).
b(n)=n+floor((n^3+1/2)^(1/2)) (A186146).
EXAMPLE
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,...)
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}]
CROSSREFS
Cf. A186146.
Sequence in context: A336497 A116883 A256543 * A364058 A335740 A352873
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 13 2011
STATUS
approved