A278101 -id:A278101 - cp's OEIS frontend

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

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 *)

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

Original entry on oeis.org

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

Jason Kimberley, Nov 15 2016

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

			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;

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..10001 (the first 3304 rows of the triangle)

Cf. A005117, A278101, A278102, A278104.

A277647:=func;
A278101_row:=funcA277647(n,k):k in[1..n^2]|IsSquarefree(k)]>;
A278103_row:=funcA278101_row(n) >;
&cat[A278103_row(n):n in[1..23]];

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 *)

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

A278115 Triangle T(n,k) = A278113(n,k)^2 A000040(k) for 1 <= k <= A278114(n), read by rows.

Original entry on oeis.org

2, 8, 3, 5, 7, 18, 12, 5, 7, 11, 13, 17, 32, 27, 20, 28, 11, 13, 17, 19, 23, 29, 31, 50, 48, 45, 28, 44, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 72, 48, 45, 63, 44, 52, 68, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 98, 75, 80, 63, 44, 52, 68, 76, 92, 29, 31, 37, 41, 43, 47, 53

Offset: 1

Jason Kimberley, Feb 10 2017

			The first six rows are:
2;
8, 3, 5, 7;
18, 12, 5, 7, 11, 13, 17;
32, 27, 20, 28, 11, 13, 17, 19, 23, 29, 31;
50, 48, 45, 28, 44, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47;
72, 48, 45, 63, 44, 52, 68, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71;

Jason Kimberley, Table of i, a(i) for i = 1..10126 (T(n,k) for n = 1..46)

Cf. A278101.

A278112:=func;
A278115_row:=funcA278112(n,p)^2*p:p in PrimesUpTo(2*n^2)]>;
&cat[A278115_row(n):n in[1..7]];

Table[# Floor[n Sqrt[2/#]]^2 &@ Prime@ k, {n, 7}, {k, PrimePi[2 n^2]}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

T(n,k) = prime(k) * floor(n*sqrt(2/prime(k)))^2.

A278100 Number of squarefree positive integers less than n^2.

Original entry on oeis.org

0, 3, 6, 11, 16, 23, 31, 39, 50, 61, 75, 89, 103, 120, 139, 157, 177, 199, 219, 243, 269, 297, 323, 351, 381, 412, 444, 477, 513, 547, 584, 624, 660, 703, 745, 789, 835, 882, 928, 977, 1025, 1073, 1124, 1174, 1230, 1285, 1342, 1400, 1460, 1523, 1582, 1645, 1708

Offset: 1

Jason Kimberley, Nov 12 2016

Jason Kimberley, Table of n, a(n) for n = 1..5000

Cf. A005117, A013928, A059956.

This is the row length sequence of A277648 and A278101.

A278100:=func;
[A278100(n):n in[1..53]]; // in cubic time

Table[Count[Range[n^2], k_ /; SquareFreeQ@ k], {n, 53}] (* Michael De Vlieger, Nov 24 2016 *)
Module[{nn=60,sf},sf=Accumulate[Table[If[SquareFreeQ[n],1,0],{n,0,nn^2}]];Table[sf[[k^2]],{k,nn}]] (* Harvey P. Dale, Nov 14 2020 *)

a(n) = #select(x->issquarefree(x), vector(n^2-1, k, k)); \\ Michel Marcus, Nov 12 2016

a(n) = A013928(n^2).

a(n) ~ 6*n^2/Pi^2 + O(n). - Amiram Eldar, Mar 09 2021

cp's OEIS Frontend

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

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

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A278115 Triangle T(n,k) = A278113(n,k)^2 A000040(k) for 1 <= k <= A278114(n), read by rows.

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A278100 Number of squarefree positive integers less than n^2.

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