A278116 -id:A278116 - cp's OEIS frontend

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

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))).

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 *)

A278119 a(n) is the index of the first occurrence of n in A278116.

Original entry on oeis.org

1, 2, 3, 5, 9, 27, 81, 163, 3451, 7372, 82177, 3716, 2769544, 25899347, 357567610, 4229297109

Offset: 1

Jason Kimberley, Feb 15 2017

a(18) = 42945966786, but a(17) > 5*10^10.

R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 106, Issue August 2018, p. 185-207.

Cf. A278106, A278116.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

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

A278119 a(n) is the index of the first occurrence of n in A278116.

Views

Author

Keywords

Comments

Links

Crossrefs