A278103 -id:A278103 - cp's OEIS frontend

A278101 Triangle T(n,k) = A277648(n,k)^2 * A005117(k), read by rows.

1, 4, 2, 3, 9, 8, 3, 5, 6, 7, 16, 8, 12, 5, 6, 7, 10, 11, 13, 14, 15, 25, 18, 12, 20, 24, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 36, 32, 27, 20, 24, 28, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 49, 32, 48, 45, 24, 28, 40, 44, 13, 14, 15, 17, 19, 21, 22

Offset: 1

Jason Kimberley, Nov 15 2016

Other that the first (with length 1), row n has length A278100(n).

Equivalently, the surd sqrt(T(n,k)) = A277648(n,k) * sqrt(A005117(k)).

			The first five rows are:
1;
4,  2,  3;
9,  8,  3,  5,  6,  7;
16,  8, 12,  5,  6,  7, 10, 11, 13, 14, 15;
25, 18, 12, 20, 24,  7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23;

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

Jason Kimberley, Table of n, a(n) for n = 1..10716 (the first 37 rows of the triangle)

Cf. A278103.

A277647:=func;
A278101_row:=funcA277647(n,j):j in[1..n^2]|IsSquarefree(j)]>;
&cat[A278101_row(n):n in[1..8]];

DeleteCases[#, 0] & /@ Table[Boole[SquareFreeQ@ k] k Floor[n/Sqrt@ k]^2, {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

A278102 a(n) is the largest j such that A278101(n,k) strictly decreases for k=1..j.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 2, 4, 2, 3, 2, 3, 4, 2, 4, 2, 3, 2, 3, 4, 2, 5, 6, 5, 2, 2, 3, 2, 4, 4, 4, 2, 2, 3, 2, 3, 4, 4, 5, 2, 2, 2, 3, 5, 3, 5, 2, 2, 2, 3, 5, 2, 4, 4, 4, 2, 3, 4, 2, 4, 5, 4, 2, 3, 2, 2, 4, 5, 4, 3, 3, 2, 2, 3, 5, 4, 5, 2, 2, 2, 3, 2, 3, 4, 2, 2, 2, 3, 2, 3, 4, 6, 5, 2, 3, 2, 2, 4, 6, 6, 2, 3, 2

Offset: 1

Jason Kimberley, Nov 15 2016

Jason Kimberley, Table of n, a(n) for n = 1..10000

This is the row length sequence for triangles A278103 and A278104.

A278106 lists first occurrences in this sequence.

A277647:=func;
A278101_row:=funcA277647(n,k):k in[1..n^2]|IsSquarefree(k)]>;
A278102:=funcA278101_row(n) >;
[A278102(n):n in[1..103]];

Map[Length@ 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 *)

A278117 Irregular triangle T(n,k) = A278115(n,k) for 1 <= k <= A278116(n), read by rows.

Original entry on oeis.org

2, 8, 3, 18, 12, 5, 32, 27, 20, 50, 48, 45, 28, 72, 48, 45, 98, 75, 128, 108, 162, 147, 125, 112, 99, 200, 192, 180, 175, 242, 192, 180, 175, 288, 243, 338, 300, 392, 363, 320, 450, 432, 405, 512, 507, 500, 448, 396, 578, 507, 500, 648, 588, 722, 675, 800, 768, 720

Offset: 1

Jason Kimberley, Feb 12 2017

Each row is the longest strictly decreasing prefix of the corresponding row of A278115.

			The first six rows are:
2;
8, 3;
18, 12, 5;
32, 27, 20;
50, 48, 45, 28;
72, 48, 45;

Jason Kimberley, Table of i, a(i) for i = 1..10104 (T(n,k) for n = 1..3333)

Cf. A278103, A278118.

A278112:=func;
A278115_row:=funcA278112(n,p)^2*p:p in PrimesUpTo(2*n^2)]>;
A278117_row:=funcA278115_row(n) >;
&cat[A278117_row(n):n in[1..20]];

Map[Take[#, 1 + Length@ TakeWhile[Differences@ #, # < 0 &]] &, #] &@ Table[# Floor[n Sqrt[2/#]]^2 &@ Prime@ k, {n, 20}, {k, PrimePi[2 n^2]}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

cp's OEIS Frontend

A278101 Triangle T(n,k) = A277648(n,k)^2 * A005117(k), read by rows.

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A278102 a(n) is the largest j such that A278101(n,k) strictly decreases for k=1..j.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A278117 Irregular triangle T(n,k) = A278115(n,k) for 1 <= k <= A278116(n), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica