A278103 Irregular triangle T(n,k) := A278101(n,k) for k = 1..A278102(n), read by rows.

1, 4, 2, 9, 8, 3, 16, 8, 25, 18, 12, 36, 32, 27, 20, 49, 32, 64, 50, 48, 45, 81, 72, 100, 98, 75, 121, 98, 144, 128, 108, 169, 162, 147, 125, 196, 162, 225, 200, 192, 180, 256, 242, 289, 288, 243, 324, 288, 361, 338, 300, 400, 392, 363, 320, 441, 392, 484, 450, 432

Offset

1,2

Comments

Each row is the longest strictly decreasing prefix of the corresponding row of A278101.

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)

Formula

T(n,k) = A278104(n,k) * A005117(k) where this triangle and A278104 both have row length sequence A278102.

Example

The first 23 rows are:
1;
4, 2;
9, 8, 3;
16, 8;
25, 18, 12;
36, 32, 27, 20;
49, 32;
64, 50, 48, 45;
81, 72;
100, 98, 75;
121, 98;
144, 128, 108;
169, 162, 147, 125;
196, 162;
225, 200, 192, 180;
256, 242;
289, 288, 243;
324, 288;
361, 338, 300;
400, 392, 363, 320;
441, 392;
484, 450, 432, 405, 384;
529, 512, 507, 500, 486, 448;

Mathematica

Map[TakeWhile[FoldList[Function[s, Boole[s < 0] #2][#2 - #1] &, #], # > 0 &] &, #] &@ Map[DeleteCases[#, 0] &, Table[Boole[SquareFreeQ@ k] k Floor[n/Sqrt@ k]^2, {n, 23}, {k, n^2}] ] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

Prog

(Magma)
A277647:=func<n, k|Isqrt(n^2 div k)>;
A278101_row:=func<n|[a^2*k where a is A277647(n, k):k in[1..n^2]|IsSquarefree(k)]>;
A278103_row:=func<n|(exists(dec){row[1..j]:j in[1..#row-1]|row[j]le row[j+1]}select dec else[1]) where row is A278101_row(n) >;
&cat[A278103_row(n):n in[1..23]];

Crossrefs

Cf. A005117, A278101, A278102, A278104.

Sequence in context: A365255 A243968 A104583 * A249327 A182728 A290538

Adjacent sequences: A278100 A278101 A278102 * A278104 A278105 A278106

Keyword

nonn,tabf,easy

Author

Jason Kimberley, Nov 15 2016

Status

approved