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A278104
Irregular triangle T(n,k) := A277648(n,k) for k = 1...A278102(n), read by rows.
4
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.
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
KEYWORD
nonn,tabf,easy
AUTHOR
Jason Kimberley, Nov 15 2016
STATUS
approved