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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.
2
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
1,2
COMMENTS
See A186145 for a discussion of adjusted joint rank sequences.
FORMULA
a(n)=n+floor(((n^3-1/2)/2)^(1/2)), A186156.
b(n)=n+floor((2n^2+1/2)^(1/3)), A186157.
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 13 2011
STATUS
approved