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

1,2

Comments

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

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)

Example

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;

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 *)

Prog

(Magma)
A277647:=func<n, k|Isqrt(n^2 div k)>;
A277648_row:=func<n|[A277647(n, k):k in[1..n^2]|IsSquarefree(k)]>;
A278101_row:=func<n|[A277647(n, k)^2*k:k in[1..n^2]|IsSquarefree(k)]>;
A278104_row:=func<n|(exists(dec){A277648_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]];

Crossrefs

Sequence in context: A337632 A057058 A334441 * A141672 A141671 A309596

Adjacent sequences: A278101 A278102 A278103 * A278105 A278106 A278107

Keyword

nonn,tabf,easy

Author

Jason Kimberley, Nov 15 2016

Status

approved