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.

User: ,9,

,9,'s wiki page.

,9, has authored 4 sequences.

A292766 Numbers n whose trajectory under iteration of the map k -> (sigma(k)+phi(k))/2 consists only of integers and is unbounded, excluding numbers n whose trajectory merges with the trajectory of a smaller number.

Original entry on oeis.org

270, 440, 496, 702, 737, 813, 828, 897, 905, 1027, 1066, 1099, 1240, 1241, 1260, 1331, 1353, 1368, 1371, 1422, 1507, 1537, 1754, 1760, 1834, 1848, 2002, 2016, 2282
Offset: 1

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Author

N. J. A. Sloane, Sep 27 2017, based on emails from Sean A. Irvine, Sep 14 2017, who computed a(1)-a(9), and Hans Havermann, same date, who computed a(10)-a(29). Hugo Pfoertner also computed many of these terms

Keywords

Comments

These are the "seeds" in A291790, that is, every number which blows up under iteration of the map k -> (sigma(k)+phi(k))/2 belongs to one of these trajectories. AT PRESENT ALL TERMS ARE CONJECTURAL.
The trajectories of these numbers are pairwise disjoint for the first 400 steps.
This is unsatisfactory because it is possible that, at some later step, these trajectories may merge, reach a prime (a fixed point), or reach a fraction (and die). However, this seems unlikely on probabilistic grounds - see the remarks of Andrew R. Booker in A292108.
Normally such a sequence would not be included in the OEIS, but exceptions have been made for this and A291790 because a number of people have worked on them, and also in the hope that this will encourage resolution of some of the open questions.
Needs a b-file.

Links

Crossrefs

Cf. A291790, A292108.

A177479 Number of permutations of 1..n avoiding adjacent step pattern up, down, down.

Original entry on oeis.org

1, 1, 2, 6, 21, 90, 450, 2619, 17334, 129114, 1067661, 9713682, 96393726, 1036348587, 11998603710, 148842430470, 1969461102357, 27688474234602, 412166988789642, 6476330295597051, 107117619952992966, 1860296912926495938, 33845967939906741213, 643778989807702357314
Offset: 0

Views

Author

Submitted independently by Signy Olafsdottir (signy06(AT)ru.is), May 09 2010 (9 terms) and R. H. Hardin, May 10 2010 (17 terms)

Keywords

Comments

Suppose j

Links

Crossrefs

Column k=0 of A242819.
Columns k=4,6 of A242784.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
       add(b(u+j-1, o-j, 1), j=1..o)+ `if`(t<2,
       add(b(u-j, o+j-1, `if`(t=0, 0, 2)), j=1..u), 0))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 07 2013
  • Mathematica
    FullSimplify[Rest[CoefficientList[Series[(E^(3*x/2) + 2*Cos[Sqrt[3]*x/2]) / (3*Cos[Sqrt[3]*x/2] - Sqrt[3]*Sin[Sqrt[3]*x/2]), {x, 0, 20}], x] * Range[0, 20]!]] (* Vaclav Kotesovec, Aug 23 2014 *)

Formula

E.g.f.: (exp(3*x/2) + 2*cos(sqrt(3)*x/2)) / (3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)). - Vaclav Kotesovec, Aug 23 2014
a(n) ~ n! * (1+exp(Pi/sqrt(3))) * 3^(3*n/2+1/2) / (2*Pi)^(n+1). - Vaclav Kotesovec, Aug 23 2014

Extensions

a(18)-a(23) from Alois P. Heinz, Oct 06 2013

A177477 Number of permutations of 1..n avoiding adjacent step pattern up, down, up.

Original entry on oeis.org

1, 1, 2, 6, 19, 70, 331, 1863, 11637, 81110, 635550, 5495339, 51590494, 524043395, 5743546943, 67478821537, 844983073638, 11240221721390, 158365579448315, 2355375055596386, 36870671943986643, 606008531691619131, 10435226671431973345, 187860338952519968538
Offset: 0

Views

Author

Submitted independently by Signy Olafsdottir (signy06(AT)ru.is), May 09 2010 (9 terms) and R. H. Hardin, May 10 2010 (17 terms)

Keywords

Comments

Suppose a < b, c < b, and c < d. To avoid abcd means not to have four consecutive letters such that the first letter is less than the second one, the third letter is less than the second one, and the third letter is less than the last one.

Links

Crossrefs

Column k=0 of A227884.
Column k=5 of A242784.
Cf. A227883, A245758.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
       add(b(u-j, o+j-1, [1, 3, 1][t]), j=1..u)+
      `if`(t=3, 0, add(b(u+j-1, o-j, 2), j=1..o)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 10 2020
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
     Sum[b[u - j, o + j - 1, {1, 3, 1}[[t]]], {j, 1, u}] +
     If[t == 3, 0, Sum[b[u + j - 1, o - j, 2], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 08 2022, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n!, where d = A245758 = 0.7827041801715217018447074977..., c = 2.035127405829990832658061124449458067... . - Vaclav Kotesovec, Aug 22 2014

Extensions

a(18)-a(23) from Alois P. Heinz, Oct 06 2013
a(0)=1 prepended by Alois P. Heinz, Mar 10 2020

A061971 Numbers n such that n divides the (left) concatenation of all numbers <= n written in base 18 (most significant digit on right).

Original entry on oeis.org

1, 17, 89, 6343, 9707, 9799, 91069, 120581, 484307, 725341, 2571749, 3432583, 4482203
Offset: 1

Views

Author

Larry Reeves (larryr(AT)acm.org), May 24 2001 a(9)-a(13) from

Keywords

Comments

This sequence differs from A029535 in that all least significant zeros are kept during concatenation.
No more terms < 8*10^6.

Examples

			See A061955 for example.

Links

Crossrefs

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Programs

  • Mathematica
    b = 18; c = {}; Select[Range[10^4], Divisible[FromDigits[c = Join[Reverse[IntegerDigits[#, b]], c], b], #] &] (* Robert Price, Mar 09 2020 *)

Extensions

Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12 2002
a(9)-a(13) from Lars Blomberg, Aug 02 2011

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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