A243496 -id:A243496 - cp's OEIS frontend

A243497 A243496 sorted into ascending order, with duplicates removed.

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 49, 53, 59, 64, 67, 81, 83, 97, 103, 121, 125, 127, 128, 131, 169, 227, 241, 243, 256, 277, 289, 311, 331, 343, 361, 419, 431, 509, 512, 529, 563, 661, 691, 709, 719, 739, 841, 961, 1009, 1024, 1433, 1523, 1619, 1681, 1787, 1849, 1879, 2063

Offset: 0

Antti Karttunen, Jun 07 2014

nonn

Matula-codes for trees which are almost "uniform", but which allow cases like 169, 841, 1009, 1681, 1849, ... where there is a special relation between prime index and the exponent. (Cf. the comments at A243496).
Differs from A214577 for the first time at n=31, where A214577(31)=227, while here we have 169 at that position, because it corresponds exactly to that "dual" case mentioned in A057546, in excess to those mentioned in A003238. Note that 169 = 13*13 = p_{2*3}^2.
a(0) = 1 stands for the empty tree.

Cf. A214577 (a subsequence), A209638, A243494.

A127301 Matula-Goebel signatures for plane general trees encoded by A014486.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 6, 7, 5, 16, 12, 12, 14, 10, 12, 9, 14, 19, 13, 10, 13, 17, 11, 32, 24, 24, 28, 20, 24, 18, 28, 38, 26, 20, 26, 34, 22, 24, 18, 18, 21, 15, 28, 21, 38, 53, 37, 26, 37, 43, 29, 20, 15, 26, 37, 23, 34, 43, 67, 41, 22, 29, 41, 59, 31, 64, 48, 48, 56, 40, 48, 36

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This sequence maps A000108(n) oriented (plane) rooted general trees encoded in range [A014137(n-1)..A014138(n)] of A014486 to A000081(n+1) distinct non-oriented rooted general trees, encoded by their Matula-Goebel numbers. The latter encoding is explained in A061773.
A005517 and A005518 give the minimum and maximum value occurring in each such range.
Primes occur at positions given by A057548 (not in order, and with duplicates), and similarly, semiprimes, A001358, occur at positions given by A057518, and in general, A001222(a(n)) = A057515(n).
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127301(SP(n)) = A127301(n) for all n, then it preserves the non-oriented form of a general tree, which implies also that it is Łukasiewicz-word permuting, satisfying A129593(SP(n)) = A129593(n) for all n >= 0. Examples of such automorphisms include A072796, A057508, A057509/A057510, A057511/A057512, A057164, A127285/A127286 and A127287/A127288.
A206487(n) tells how many times n occurs in this sequence. - Antti Karttunen, Jan 03 2013

			A000081(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, A014486(5) = 44 (= 101100 in binary = A063171(5)), encodes the following plane tree:
.....o
.....|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(1) * A000040(A000040(1)) = 2*3 = 6, thus a(5)=6.
Likewise, A014486(6) = 50 (= 110010 in binary = A063171(6)) encodes the plane tree:
.o
.|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(A000040(1)) * A000040(1) = 3*2 = 6, thus a(6) is also 6, which shows these two trees are identical if one ignores their orientation.

Antti Karttunen, Table of n, a(n) for n = 0..6917
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz
Index entries for sequences related to Matula-Goebel numbers

a(A014138(n)) = A007097(n+1), a(A014137(n)) = A000079(n+1) for all n.
a(|A106191(n)|) = A033844(n-1) for all n >= 1.
Cf. A001222, A005517, A005518, A057515, A057518, A057548, A127302, A129593, A153826, A209638, A243491, A243492, A243494, A243496.
For standard instead of binary encoding we have A358506.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists binary encodings of ordered rooted trees.
Cf. A001263, A061775, A196050, A206487, A358505.

mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
binbalQ[n_]:=n==0||With[{dig=IntegerDigits[n,2]},And@@Table[If[k==Length[dig],SameQ,LessEqual][Count[Take[dig,k],0],Count[Take[dig,k],1]],{k,Length[dig]}]];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[mgnum[bint[n]],{n,Select[Range[0,1000],binbalQ]}] (* Gus Wiseman, Nov 22 2022 *)

(define (A127301 n) (*A127301 (A014486->parenthesization (A014486 n)))) ;; A014486->parenthesization given in A014486.
(define (*A127301 s) (if (null? s) 1 (fold-left (lambda (m t) (* m (A000040 (*A127301 t)))) 1 s)))

A001222(a(n)) = A057515(n) for all n.

A243493 Value of Matula-Goebel signature at the fixed points of A069787: a(n) = A127301(A243490(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 5, 16, 12, 10, 14, 13, 11, 32, 24, 20, 28, 26, 22, 37, 23, 34, 31, 64, 48, 40, 56, 52, 44, 74, 46, 68, 62, 76, 39, 89, 61, 47, 86, 101, 118, 109, 127, 128, 96, 80, 112, 104, 88, 148, 92, 136, 124, 152, 78, 178, 122, 94, 172, 202, 236, 218, 254

Offset: 0

Antti Karttunen, Jun 07 2014

The first duplicate value occurs at n=101, as a(101) = a(129) = 362. The corresponding A014486-indices are A243490(101) = 924 and A243490(129) = 1640, respectively.

Antti Karttunen, Table of n, a(n) for n = 0..525

A243494 gives the same terms sorted into ascending order with duplicates removed.

Cf. A243496, A209636, A209637, A069787, A127301, A243490, A243491, A000079, A007097, A036256.

a(n) = A127301(A243490(n)).
a(n) = A243491(A243490(n)).
For all n >= 0, a(A036256(n)-1) = A007097(n) and a(A036256(n)) = A000079(n+1).

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A243497 A243496 sorted into ascending order, with duplicates removed.

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A127301 Matula-Goebel signatures for plane general trees encoded by A014486.

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A243493 Value of Matula-Goebel signature at the fixed points of A069787: a(n) = A127301(A243490(n)).

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A243495 Indices in A014486 for the oriented trees that stay the same when "deep-rotated": fixed points of A057511/A057512.

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