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 1-9 of 9 results.

A081165 Largest prime factor of A060114.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 5, 5, 7, 11, 13, 89, 131, 479, 479, 1879, 2153, 3167, 4463
Offset: 0

Views

Author

Antti Karttunen and Wouter Meeussen, Mar 10 2003

Keywords

Crossrefs

Cf. A006530, A060114, A081164, A081166.

Formula

a(n) = A006530(A060114(n)).

A081166 Number of distinct primes dividing A060114(n).

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 3, 3, 4, 5, 6, 12, 16, 20, 31, 50, 70, 106, 143
Offset: 0

Views

Author

Antti Karttunen and Wouter Meeussen, Mar 10 2003

Keywords

Crossrefs

Cf. A001221, A060114, A081165.

Formula

a(n) = A001221(A060114(n)).

A060116 First quotients of A060114.

Original entry on oeis.org

1, 2, 3, 1, 5, 4, 6, 21, 77, 13, 18759792910, 51104148, 226096622, 285439624304460249957765, 9476683358538538610273746404006394881428593545
Offset: 0

Views

Author

Antti Karttunen, Mar 01 2001

Keywords

Links

Programs

Extensions

a(8)-a(14) from Sean A. Irvine, Oct 25 2022

A057505 Signature-permutation of a Catalan Automorphism: Donaghey's map M acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Comments

This is equivalent to map M given by Donaghey on page 81 of his paper "Automorphisms on ..." and also equivalent to the transformation procedure depicted in the picture (23) of Donaghey-Shapiro paper.
This can be also considered as a "more recursive" variant of A057501 or A057503 or A057161.

References

  • D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees--History of Combinatorial Generation, vi+120pp. ISBN 0-321-33570-8 Addison-Wesley Professional; 1ST edition (Feb 06, 2006).

Links

Crossrefs

Inverse: A057506.
The 2nd, 3rd, 4th, 5th and 6th "power": A071661, A071663, A071665, A071667, A071669.
Other related permutations: A057501, A057503, A057161.
Cycle counts: A057507. Maximum cycle lengths: A057545. LCM's of all cycles: A060114. See A057501 for other Maple procedures.
Row 17 of table A122288.
Cf. A080981 (the "primitive elements" of this automorphism), A079438, A079440, A079442, A079444, A080967, A080968, A080972, A080272, A080292, A083929, A080973, A081164, A123050, A125977, A126312.

Programs

  • Maple
    map(CatalanRankGlobal,map(DonagheysM, A014486)); or map(CatalanRankGlobal,map(DeepRotateTriangularization, A014486));
DonagheysM := n -> pars2binexp(DonagheysMP(binexp2pars(n)));
DonagheysMP := h -> `if`((0 = nops(h)),h,[op(DonagheysMP(car(h))),DonagheysMP(cdr(h))]);
DeepRotateTriangularization := proc(nn) local n,s,z,w; n := binrev(nn); z := 0; w := 0; while(1 = (n mod 2)) do s := DeepRotateTriangularization(BinTreeRightBranch(n))*2; z := z + (2^w)*s; w := w + binwidth(s); z := z + (2^w); w := w + 1; n := floor(n/2); od; RETURN(z); end;

Formula

a(0) = 0, and for n>=1, a(n) = A085201(a(A072771(n)), A057548(a(A072772(n)))). [This recurrence reflects the S-expression implementation given first in the Program section: A085201 is a 2-ary function corresponding to 'append', A072771 and A072772 correspond to 'car' and 'cdr' (known also as first/rest or head/tail in some languages), and A057548 corresponds to unary form of function 'list'].
As a composition of related permutations:
a(n) = A057164(A057163(n)).
a(n) = A057163(A057506(A057163(n))).

A057507 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Original entry on oeis.org

1, 1, 1, 2, 3, 10, 18, 46, 95, 236, 528, 1288, 2936, 6984, 16212, 38528, 90717, 216648, 516358, 1240818, 2979992
Offset: 0

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Comments

For the convenience of the range notation above, we define A014137(-1) and A014138(-1) as zero.

Links

Crossrefs

a(n) = A081148(n)+A081150(n). Bisections: A081151, A081167. Cf. A057545, A060114, A081164.
Occurs for first time in A073201 as row 2614.

A057545 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Original entry on oeis.org

1, 1, 2, 3, 6, 6, 24, 72, 144, 147, 588, 672, 2136, 10152, 11520, 29484, 117936, 270576, 656352, 2062368, 4040160
Offset: 0

Views

Author

Antti Karttunen, Sep 07 2000

Keywords

Comments

For the convenience of the range notation above, we define A014137(-1) and A014138(-1) as zero.
Equal to the degree of the polynomials M_n(x) Donaghey gives on the page 81 of his paper.
Factored terms: 1, 1, 2, 3, 2*3, 2*3, 2^3 * 3, 2^3 * 3^2, 2^4 * 3^2, 3 * 7^2, 2^2 * 3 * 7^2, 2^5 * 3 * 7, 2^3 * 3 * 89, 2^3 * 3^3 * 47, 2^8 * 3^2 * 5, 2^2 * 3^4 * 7 * 13, 2^4 * 3^4 * 7 * 13, 2^4 * 3^2 * 1879, 2^5 * 3^2 * 43 * 53, 2^5 * 3^3 * 7 * 11 * 31, 2^5 * 3 * 5 * 19 * 443

Links

Crossrefs

Cf. A057507, A060114, A080967, A081165.
Occurs for first time in A073203 as row 2614.

A060113 Least common multiple of all orbit lengths of the permutation A038776.

Original entry on oeis.org

1, 1, 1, 3, 12, 48, 1392, 214402800, 3817990510765200, 4738197524832401740110000, 1091118722532825192362856035208963090000, 2182237445065650384725712070417926180000, 507772214574105904772762129719909048054921244582661534399180000
Offset: 0

Views

Author

Antti Karttunen, Mar 01 2001

Keywords

Links

Crossrefs

Cf. A060114, A060115.

Extensions

More terms from Sean A. Irvine, Oct 25 2022

A073204 Array of LCMs-of-cycle-lengths sequences for the table A073200.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 6, 2, 1, 1, 2, 2, 2, 12, 2, 2, 1, 1, 2, 2, 2, 120, 2, 6, 2, 1, 1, 2, 2, 2, 120, 2, 12, 4, 1, 1, 1, 2, 2, 2, 840, 2, 120, 8, 1, 2, 1, 1, 2, 2, 2, 840, 2, 120, 16, 1, 4, 1, 1, 1, 2, 2, 2, 5040, 2, 840, 32, 1, 8, 2, 2, 1, 1
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

Each row of this table gives the least common multiple of all cycle lengths produced by the Catalan bijection (given in the corresponding row of A073200) when it acts on A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Links

  • A. Karttunen, Gatomorphisms (With the complete source and explanation)

Crossrefs

Cf. also A073201-A073203.
Few EIS-sequences occur in this table. The first known occurrences are: rows 6 and 8: A011782, Row 7: A000012, Row 2614: A060114, Row 2618 (?), ..., 17517: A057544.

A081164 Number of distinct cycle lengths in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 11, 15, 25, 32, 64, 88, 155, 234, 423, 647, 1184, 1800
Offset: 0

Views

Author

Wouter Meeussen and Antti Karttunen, Mar 10 2003

Keywords

Comments

This is the number of nonzero, non-constant terms of the polynomials M_n(x) Donaghey gives on the page 81 of his paper. The term x^18 seems to have been accidentally dropped from the polynomial M_7(x).

Links

Crossrefs

Cf. A081166, A060114.
Showing 1-9 of 9 results.

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