cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A278113 Triangle T(n,k) = A278112(n,A000040(k)) for 1 <= k <= A278114(n), read by rows.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 4, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Jason Kimberley, Feb 09 2017

Keywords

Comments

This triangle consists of those columns of A278112 that have prime index.

Examples

			The first eight rows are:
  1;
  2, 1, 1, 1;
  3, 2, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  7, 5, 4, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Links

Crossrefs

Cf. A000040, A277648, A278112, A278114, A278115, A278118.

Programs

  • Magma
    A278112:=func;
A278113_row:=funcA278112(n,p):p in PrimesUpTo(2*n^2)]>;
&cat[A278113_row(n):n in[1..8]];
  • Mathematica
    Table[Floor[n Sqrt[2/Prime@ k]], {n, 8}, {k, PrimePi[2 n^2]}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

Formula

T(n,k) = floor(n*sqrt(2/prime(k))).
T(n,k) sqrt(A000040(k)) <= n sqrt(2) < (T(n,k)+1) sqrt(A000040(k)).

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

Views

Author

Jason Kimberley, Feb 10 2017

Keywords

Examples

			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;

Links

Crossrefs

Cf. A278101.

Programs

  • Magma
    A278112:=func;
A278115_row:=funcA278112(n,p)^2*p:p in PrimesUpTo(2*n^2)]>;
&cat[A278115_row(n):n in[1..7]];
  • Mathematica
    Table[# Floor[n Sqrt[2/#]]^2 &@ Prime@ k, {n, 7}, {k, PrimePi[2 n^2]}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

Formula

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

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

Views

Author

Jason Kimberley, Feb 12 2017

Keywords

Comments

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

Examples

			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;

References

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

Links

Crossrefs

Cf. A278104.

Programs

  • Magma
    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]];
  • Mathematica
    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 *)

Formula

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

Views

Author

Jason Kimberley, Feb 12 2017

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A278103, A278118.

Programs

  • Magma
    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]];
  • Mathematica
    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 *)
Showing 1-4 of 4 results.

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