A278104 -id:A278104 - 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.

A278118 Irregular triangle T(n,k) = A278113(n,k) for 1 <= k <= A278116(n), read by rows.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 5, 4, 3, 2, 6, 4, 3, 7, 5, 8, 6, 9, 7, 5, 4, 3, 10, 8, 6, 5, 11, 8, 6, 5, 12, 9, 13, 10, 14, 11, 8, 15, 12, 9, 16, 13, 10, 8, 6, 17, 13, 10, 18, 14, 19, 15, 20, 16, 12, 10, 21, 17, 13, 22, 17, 13, 23, 18, 24, 19, 25, 20, 15, 26, 21, 16, 13, 27, 22, 17, 14, 11, 10

Offset: 1

Jason Kimberley, Feb 12 2017

This triangle lists the "descending sequences for rank 1" of Eggleton et al.

			For example, 6 sqrt(2) > 4 sqrt(3) > 3 sqrt(5), because 72 > 48 > 45.
The first six rows are:
1;
2, 1;
3, 2, 1;
4, 3, 2;
5, 4, 3, 2;
6, 4, 3;

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

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

Cf. A278104.

A278112:=func;
A278115_row:=funcA278112(n,p)^2*p:p in PrimesUpTo(2*n^2)]>;
A278116:=funcA278115_row(n)>;
A278118_row:=funcA278112(n,NthPrime(k)):k in[1..A278116(n)]]>;
[A278118_row(n):n in[1..20]];

Function[w, MapIndexed[Take[w[[First@ #2, 1]], 1 + Length@ TakeWhile[ Differences@ #1, # < 0 &]] &, w[[All, -1]]]]@ Table[Function[k, Function[p, {#, p #^2} &@ Floor[n Sqrt[2/p]]]@ Prime@ k]@ Range@ PrimePi[2 n^2], {n, 27}] (* Michael De Vlieger, Feb 17 2017 *)

From A278113: T(n,k) sqrt(prime(k)) <= n sqrt(2) < (T(n,k)+1) sqrt(prime(k)).
Here, we also have:
T(n,1) sqrt(2) > T(n,2) sqrt(3) > ... > T(n,A278116(n)) sqrt(prime(A278116(n))).

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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A278118 Irregular triangle T(n,k) = A278113(n,k) for 1 <= k <= A278116(n), read by rows.

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