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 1-10 of 10 results.

A130924 Signature permutation of a Catalan automorphism: Inverse KROF-transform of automorphism *A120706.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 4, 5, 17, 18, 15, 14, 16, 20, 19, 21, 9, 10, 22, 11, 12, 13, 48, 46, 50, 49, 45, 40, 39, 41, 37, 38, 43, 42, 44, 47, 54, 55, 52, 51, 53, 57, 56, 58, 23, 24, 59, 25, 26, 27, 61, 60, 62, 28, 29, 63, 30, 31, 32, 64, 33, 34, 35, 36, 141, 145, 134, 133, 139
Offset: 0

Views

Author

Antti Karttunen, Jun 11 2007

Keywords

Comments

This is the unique Catalan automorphism f, such that *A120706 = (KROF f). See A122202 for the definition of KROF.

Links

Crossrefs

Inverse: A130923. Cf. A130925 & A130926.

A130925 Signature permutation of a Catalan automorphism: Inverse FORK-transform of automorphism *A120706.

Original entry on oeis.org

0, 1, 3, 2, 6, 8, 7, 4, 5, 14, 15, 19, 21, 22, 16, 17, 20, 9, 10, 18, 11, 12, 13, 37, 38, 39, 40, 41, 51, 52, 56, 62, 59, 60, 64, 63, 58, 42, 43, 44, 45, 46, 53, 54, 55, 23, 24, 61, 25, 26, 27, 47, 48, 57, 28, 29, 49, 30, 31, 32, 50, 33, 34, 35, 36, 107, 108, 109, 110, 111
Offset: 0

Views

Author

Antti Karttunen, Jun 11 2007

Keywords

Comments

This is the unique Catalan automorphism f, such that *A120706 = (FORK f). See A122201 for the definition of FORK.

Links

Crossrefs

Inverse: A130926. Cf. A130923 & A130924.

A120707 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A120705/A120706.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 39, 72, 138, 286, 512
Offset: 0

Views

Author

Antti Karttunen, Jun 28 2006

Keywords

Comments

The number of orbits to which the corresponding automorphisms partition the set of A000108(n) binary trees of n internal nodes.

A120708 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A120705/A120706.

Original entry on oeis.org

1, 1, 2, 3, 8, 24, 30, 60, 262, 262, 950, 2508, 4964
Offset: 0

Views

Author

Antti Karttunen, Jun 28 2006

Keywords

A120709 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A120705/A120706.

Original entry on oeis.org

1, 1, 2, 6, 8, 120, 18480, 314954640, 29650259293200, 806144692105283180937910919057361600, 116647824244848662624579303522985859096483200, 3116996669196650347010384586809853826278378997194387292312487616560888473194736151916402811882584000
Offset: 0

Views

Author

Antti Karttunen, Jun 28 2006

Keywords

A074679 Signature permutation of a Catalan automorphism: Rotate binary tree left if possible, otherwise swap its sides.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 8, 4, 5, 14, 15, 16, 17, 18, 19, 20, 21, 9, 10, 22, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 23, 24, 59, 25, 26, 27, 60, 61, 62, 28, 29, 63, 30, 31, 32, 64, 33, 34, 35, 36, 107, 108, 109, 110, 111
Offset: 0

Views

Author

Antti Karttunen, Sep 11 2002

Keywords

Comments

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)
...B...C.......A...B
....\./.........\./
.A...x....-->....x...C.................A..().........()..A..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> ((a . b) . c) ____ (a . ()) --> (() . a)
That is, we rotate the binary tree left, in case it is possible and otherwise (if the right hand side of a tree is a terminal node) swap the left and right subtree (so that the terminal node ends to the left hand side), i.e., apply the automorphism *A069770. Look at the example in A069770 to see how this will produce the given sequence of integers.
This is the first multiclause nonrecursive automorphism in table A089840 and the first one whose order is not finite, i.e., the maximum size of cycles in this permutation is not bounded (see A089842). The cycle counts in range [A014137(n-1)..A014138(n)] of this permutation is given by A001683(n+1), which is otherwise the same sequence as for Catalan automorphisms *A057161/*A057162, but shifted once right. For an explanation, please see the notes in OEIS Wiki.

Links

Crossrefs

This automorphism has several variants, where the first clause is same (rotate binary tree to the left, if possible), but something else is done (than just swapping sides), in case the right hand side is empty: A082335, A082349, A123499, A123695. The following automorphisms can be derived recursively from this one: A057502, A074681, A074683, A074685, A074687, A074690, A089865, A120706, A122321, A122332. See also somewhat similar ones: A069773, A071660, A071656, A071658, A072091, A072095, A072093.
Inverse: A074680.
Row 12 of A089840.
Occurs also in A073200 as row 557243 because a(n) = A073283(A073280(A072796(n))). a(n) = A083927(A123498(A057123(n))).
Number of cycles: LEFT(A001683). Number of fixed points: LEFT(A019590). Max. cycle size & LCM of all cycle sizes: A089410 (in range [A014137(n-1)..A014138(n)] of this permutation).

Extensions

Description clarified Oct 10 2006

A120705 Permutation of natural numbers induced by the Catalan bijection gma120705 acting on the binary trees encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 4, 5, 6, 22, 21, 17, 18, 20, 10, 9, 11, 13, 12, 14, 15, 19, 16, 64, 63, 58, 59, 62, 46, 45, 48, 50, 49, 54, 55, 61, 57, 27, 26, 23, 24, 25, 29, 28, 33, 34, 35, 30, 36, 32, 31, 38, 37, 39, 41, 40, 51, 52, 60, 56, 42, 43, 44, 47, 53, 196, 195, 189, 190, 194
Offset: 0

Views

Author

Antti Karttunen, Jun 28 2006

Keywords

Comments

When the automorphisms A120705/A120705 act on the full Stern-Brocot tree (A007305/A047679), which is an infinite binary tree, they will move each fraction r to the position of 2*r (or r/2 respectively). See comments at A065249 and A065251. (Proof in preparation, to be published.)

Links

Crossrefs

Inverse of A120706. Cf. A074680.
Number of cycles: A120707. Number of fixed-points: A019590. Max. cycle size: A120708. LCM of cycle sizes: A120709.

A130923 Signature permutation of a Catalan automorphism: Inverse FORK-transform of automorphism *A120705.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 5, 4, 6, 17, 18, 20, 21, 22, 12, 11, 13, 9, 10, 15, 14, 16, 19, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 31, 32, 29, 28, 30, 34, 33, 35, 27, 24, 36, 23, 26, 25, 40, 39, 41, 37, 38, 43, 42, 44, 47, 52, 51, 53, 56, 60, 129, 130, 132, 133, 134
Offset: 0

Views

Author

Antti Karttunen, Jun 11 2007

Keywords

Comments

This is the unique Catalan automorphism f, such that *A120705 = (FORK f). See A122201 for the definition of FORK.

Links

Crossrefs

Inverse: A130924. Cf. A130925 & A130926.

A130926 Signature permutation of a Catalan automorphism: Inverse KROF-transform of automorphism *A120705.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 4, 6, 5, 17, 18, 20, 21, 22, 9, 10, 14, 15, 19, 11, 16, 12, 13, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 23, 24, 25, 26, 27, 37, 38, 39, 40, 41, 51, 52, 56, 60, 28, 29, 42, 43, 44, 30, 53, 36, 32, 33, 47, 31, 35, 34, 129, 130, 132, 133, 134
Offset: 0

Views

Author

Antti Karttunen, Jun 11 2007

Keywords

Comments

This is the unique Catalan automorphism f, such that *A120705 = (KROF f). See A122202 for the definition of KROF.

Links

Crossrefs

Inverse: A130925. Cf. A130923 & A130924.

A253288 Each term a(n) satisfies four properties: 1, divisible by all prime factors of n; 2, divisible by only the prime factors of n; 3, not equal to any of the terms a(1), a(2), ... a(n-1); 4, smallest number satisfying 1-3 if A005361(n) is even, or second smallest number satisfying 1-3 if A005361(n) is odd.

Original entry on oeis.org

1, 4, 9, 2, 25, 12, 49, 16, 3, 20, 121, 6, 169, 28, 45, 8, 289, 18, 361, 10, 63, 44, 529, 36, 5, 52, 81, 14, 841, 60, 961, 64, 99, 68, 175, 24, 1369, 76, 117, 50, 1681, 84, 1849, 22, 15, 92, 2209, 48, 7, 40, 153, 26, 2809, 72, 275, 98, 171, 116, 3481, 30, 3721, 124, 21, 32
Offset: 1

Views

Author

N. J. A. Sloane, Dec 29 2014

Keywords

Comments

This sequence is permutation of the positive integers.
The prime p occurs at n = p^2.
Multiples of a number x have density 1/x.
Conjecture: this permutation of positive integers is self-inverse. Compare with A358971. The principal distinction between this sequence and A358971 is that fixed points aside from A358971(1) = 1 are explicitly ruled out in the latter. - Michael De Vlieger, Dec 10 2022

References

  • Brad Klee, Posting to Sequence Fans Mailing List, Dec 21, 2014.

Links

Crossrefs

Cf. A005361 (Product of exponents of prime factorization of n), A358971.

Programs

  • Maple
    A253288div := proc(a,n)
    local npr,d,apr ;
    npr := numtheory[factorset](n) ;
    for d in npr do
        if modp(a,d) <> 0 then
            return false;
        end if;
    end do:
    apr := numtheory[factorset](a) ;
    if apr minus npr = {} then
        true;
    else
        false;
    end if;
end proc:
A253288 := proc(n)
    option remember;
    local a,i,prev,act,ev ;
    if n =1 then
        1;
    else
        act := 1 ;
        if type(A005361(n),'even') then
            ev := true;
        else
            ev := false;
        end if;
        for a from 1 do
            prev := false;
            for i from 1 to n-1 do
                if procname(i) = a then
                    prev := true;
                    break;
                end if;
            end do:
            if not prev then
                if A253288div(a,n) then
                    if ev or act > 1 then
                        return a;
                    else
                        act := act+1 ;
                    end if;
                end if;
            end if;
        end do:
    end if;
end proc:
seq(A253288(n),n=1..80) ; # R. J. Mathar, Jan 22 2015
  • Mathematica
    nn = 1000; c[] = False; q[] = 1; f[n_] := f[n] = Map[Times @@ # &, Transpose@ FactorInteger[n]]; a[1] = 1; c[1] = True; u = 2; Do[Which[PrimeQ[n], k = n^2, PrimeQ@ Sqrt[n], k = Sqrt[n], SquareFreeQ[n], k = First@ f[n]; m = q[k]; While[Nand[! c[k m], k m != n, Divisible[k, First@ f[m]]], m++]; While[Nor[c[q[k] k], Divisible[k, First@ f[q[k]]]], q[k]++]; k *= m, True, t = 0; Set[{k, s}, {First[#], 1 + Boole@ OddQ@ Last[#]} &[f[n]]]; m = q[k]; Until[t == s, If[m > q[k], m++]; While[Nand[! c[k m], Divisible[k, First@f[m]]], m++]; t++]; If[s == 1, While[Nor[c[q[k] k], Divisible[k, First@ f[q[k]]]], q[k]++]]; k *= m]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Dec 10 2022 *)

Extensions

Terms beyond 361 from R. J. Mathar, Jan 22 2015
Showing 1-10 of 10 results.

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

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