A278104 Irregular triangle T(n,k) := A277648(n,k) for k = 1...A278102(n), read by rows.

1, 2, 1, 3, 2, 1, 4, 2, 5, 3, 2, 6, 4, 3, 2, 7, 4, 8, 5, 4, 3, 9, 6, 10, 7, 5, 11, 7, 12, 8, 6, 13, 9, 7, 5, 14, 9, 15, 10, 8, 6, 16, 11, 17, 12, 9, 18, 12, 19, 13, 10, 20, 14, 11, 8, 21, 14, 22, 15, 12, 9, 8, 23, 16, 13, 10, 9, 8, 24, 16, 13, 10, 9, 25, 17, 26, 18, 27, 19, 15, 28, 19

Offset: 1

Author

Jason Kimberley, Nov 15 2016

Keywords

• nonn
• tabf
• easy

Comments

This triangle lists the "descending sequences across ranks" of Eggleton et al.

Examples

			The first 23 rows are:
1;
2,  1;
3,  2,  1;
4,  2;
5,  3,  2;
6,  4,  3,  2;
7,  4;
8,  5,  4,  3;
9,  6;
10,  7,  5;
11,  7;
12,  8,  6;
13,  9,  7,  5;
14,  9;
15, 10,  8,  6;
16, 11;
17, 12,  9;
18, 12;
19, 13, 10;
20, 14, 11,  8;
21, 14;
22, 15, 12,  9,  8;
23, 16, 13, 10,  9,  8;
		

References

• R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.

Links

• Jason Kimberley, Table of n, a(n) for n = 1..10001 (the first 3304 rows of the triangle)

Programs

Magma

A277647:=func;
A277648_row:=funcA277647(n,k):k in[1..n^2]|IsSquarefree(k)]>;
A278101_row:=funcA277647(n,k)^2*k:k in[1..n^2]|IsSquarefree(k)]>;
A278104_row:=funcA277648_row(n)[1..j]:j in[1..#row-1]|row[j]le row[j+1]}select dec else[1]) where row is A278101_row(n) >;
&cat[A278104_row(n):n in[1..23]];

Mathematica

Map[Last, #, {2}] &@ Map[TakeWhile[FoldList[Function[s, Boole[s < 0] {First@ #2, Last@ #2}][First@ #2 - First@ #1] &, #], Total@ # > 0 &] &, #] &@ Map[DeleteCases[#, {0, 0}] &, Table[Boole[SquareFreeQ@ k] {k #^2, #} &@ Floor[n/Sqrt@ k], {n, 32}, {k, n^2}] ] // Flatten (* Michael De Vlieger, Nov 24 2016 *)