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.

A167906 Fixed points of permutations A121878, A167904, A167905.

Original entry on oeis.org

1, 2, 3, 4, 14, 17, 18, 19, 20, 21, 22, 35, 36, 37, 42, 44, 45, 46, 47, 48, 49, 63, 64, 65, 66, 67, 86, 89, 90, 91, 92, 93, 94, 107, 108, 109, 110, 111, 112, 123, 132, 134, 135, 136, 137, 146, 148, 161, 162, 168, 170, 171, 179, 180, 185, 186, 187, 189, 191, 192, 193
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 15 2009

Keywords

Comments

A121878(a(n))=a(n); A167904(a(n))=a(n); A167905(a(n))=a(n).

Links

Programs

  • PARI
    print1(1,", ");v=[1]; n=1; while(n<100, if(issquarefree(n+v[#v])&&!vecsearch(vecsort(v), n), v=concat(v, n);if(n==#v,print1(n,", ")); n=0); n++) \\ Derek Orr, Jun 09 2015

A121878 a(1)=1. a(n) = the smallest positive integer not occurring earlier in the sequence such that a(n-1)+a(n) is squarefree.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 12, 11, 15, 14, 16, 13, 17, 18, 19, 20, 21, 22, 24, 23, 28, 25, 26, 27, 30, 29, 32, 33, 34, 31, 35, 36, 37, 40, 38, 39, 43, 42, 41, 44, 45, 46, 47, 48, 49, 52, 50, 51, 54, 53, 56, 55, 58, 57, 61, 62, 60, 59, 63, 64, 65, 66, 67, 70, 68, 69, 72, 71
Offset: 1

Views

Author

Leroy Quet, Aug 31 2006

Keywords

Comments

Inverse: A167905; A167904(n) = a(a(n)). [Reinhard Zumkeller, Nov 15 2009]
Conjectured to be a permutation of the natural numbers. - Derek Orr, Jun 01 2015

Examples

			9,10,11,12,... are the positive integers not occurring among the first 8 terms of the sequence. a(8) + 9 = 16, which is not squarefree. a(8) + 10 = 17, which is squarefree. So a(9) = 10.

Links

Crossrefs

Cf. A167907, A075380. [Reinhard Zumkeller, Nov 15 2009]

Programs

  • Mathematica
    f[s_] := Block[{k = 1},While[MemberQ[s, k] || Max @@ Last /@ FactorInteger[(s[[ -1]] + k)] > 1, k++ ]; Append[s, k]]; Nest[f, {1}, 75] (* Ray Chandler, Sep 06 2006 *)
  • PARI
    v=[1];n=1;while(n<100,if(issquarefree(v[#v]+n)&&!vecsearch(vecsort(v),n),v=concat(v,n);n=0);n++);v \\ Derek Orr, Jun 01 2015

Extensions

Extended by Ray Chandler, Sep 06 2006

A167905 Inverse integer permutation to A121878.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 12, 11, 16, 14, 13, 15, 17, 18, 19, 20, 21, 22, 24, 23, 26, 27, 28, 25, 30, 29, 34, 31, 32, 33, 35, 36, 37, 39, 40, 38, 43, 42, 41, 44, 45, 46, 47, 48, 49, 51, 52, 50, 54, 53, 56, 55, 58, 57, 62, 61, 59, 60, 63, 64, 65, 66, 67, 69, 70, 68, 72, 71
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 15 2009

Keywords

Comments

a(A167906(n)) = A167906(n);
a(A121878(n)) = A121878(a(n)) = n;
a(A167904(n)) = A167904(a(n)) = A121878(n).

Links

A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

1, 5, 15, 45, 105, 264, 555, 1221, 2445, 4935, 9324, 17941, 32522, 59400, 104808, 184569, 315711, 540540, 902335, 1504800, 2462724, 4014513, 6444425, 10316250, 16283707, 25610886, 39841865, 61720659, 94687230, 144731706, 219282679, 330996105, 495901413, 740046425
Offset: 1

Views

Author

Olivier Gérard, May 31 2024

Keywords

Comments

The length of each row is given by A000041.
As many sequences start like the positive integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A000027 to have the same row sums for at least 8 terms.
A069782, A088480, A090107, A090108, A090109, A115510, A115511, A130446, A131717, A132086, A153671, A167904, A296879, A296882, A296885, A296888, A296891, A296894, A296897, A296900, A296903, A296906, A303502, A317945, A335280, A361374.

Examples

			Let's put the list of integers in a triangle whose rows have length p(n), number of integer partitions of n.
.
    1 |  1
    5 |  2  3
   15 |  4  5  6
   45 |  7  8  9 10 11
  105 | 12 13 14 15 16 17 18
  264 | 19 20 21 22 23 24 25 26 27 28 29
  555 | 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
.
The sequence gives the row sums of this triangle.

Crossrefs

Cf. A000027, seen as a triangle with shape A000041.
Cf. A373301, the same principle, but starting from integer zero instead of 1.
Cf. A006003, row sums of the integers but for the linear triangle.

Programs

  • Mathematica
    Module[{s = 0},
 Table[s +=
   PartitionsP[n - 1]; (s + PartitionsP[n])*(s + PartitionsP[n] - 1)/2 -
   s*(s - 1)/2, {n, 1, 30}]]
Showing 1-4 of 4 results.

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