A209637 -id:A209637 - cp's OEIS frontend

A209638 Sequence A209636 (or A209637) sorted into ascending order.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 22, 24, 26, 28, 29, 31, 32, 34, 37, 38, 40, 41, 43, 44, 48, 52, 53, 56, 58, 59, 62, 64, 67, 68, 71, 74, 76, 79, 80, 82, 86, 88, 89, 96, 101, 104, 106, 107, 109, 112, 116, 118, 124, 127, 128, 131, 134

Offset: 0

Antti Karttunen, Mar 11 2012

nonn

These are Matula-numbers (see A061773) for the rooted trees where no vertices with more than one non-leaf branch ever occur. In other words, natural numbers which are either some power of 2, or of the form 2^k * p_i, where k >= 0, and p_i is the i-th prime (A000040(i)), with i being one of the terms of this sequence.

Index entries for sequences related to Matula-Goebel numbers

Sorted version of A209636 and A209637.

Subset of A093641 and A122132.

from sympy import prime
def a(n):
    n = 2*n
    m = 1
    if n<2: return 1
    while n>1:
        if n%2==0:
            n//=2
            m*=2
        else:
            n=(n - 1)//2
            m=prime(m)
    return m
print(sorted([a(n) for n in range(101)])) # Indranil Ghosh, May 26 2017

A278542 a(n) = A046523(A209637(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 6, 8, 2, 2, 2, 2, 6, 6, 12, 16, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 6, 12, 12, 24, 32, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12, 12, 24, 24, 48, 64, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12

Offset: 0

Antti Karttunen, Nov 30 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..512

Cf. A046523, A054429, A209637, A278540, A278541.

(define (A278542 n) (A046523 (A209637 n)))

a(n) = A046523(A209637(n)).

a(n) = A278541(A054429(n)).

A209636 Matula-numbers computed for rooted trees encoded by A071162/A071163.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 7, 5, 16, 12, 14, 10, 19, 13, 17, 11, 32, 24, 28, 20, 38, 26, 34, 22, 53, 37, 43, 29, 67, 41, 59, 31, 64, 48, 56, 40, 76, 52, 68, 44, 106, 74, 86, 58, 134, 82, 118, 62, 131, 89, 107, 71, 163, 101, 139, 79, 241, 157, 191, 109, 331, 179, 277

Offset: 0

Antti Karttunen, Mar 11 2012

Sequence is injective: Any number occurs at most once, as each plane tree encoded by A071162/A071163 is mapped to a unique non-oriented rooted tree. See also A209637, A209638.

Sequence A209638 gives the same terms sorted into ascending order.

Indranil Ghosh (terms 0..1000) & Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Matula-Goebel numbers

Cf. A000040, A054429, A071162, A071163, A127301, A209637, A278541.

A209636(n) = { my(n=2*n, m=1); while(n >= 2, if(!(n%2),m*=2,m = prime(m)); n\=2); m; } \\ Antti Karttunen, May 25 2017

from sympy import prime
def a(n):
    n = 2*n
    m = 1
    if n<2: return 1
    while n>1:
        if n%2==0:
            n//=2
            m*=2
        else:
            n=(n - 1)//2
            m=prime(m)
    return m
print([a(n) for n in range(101)]) # Indranil Ghosh, May 25 2017, translated from Antti Karttunen's SCHEME code

(define (A209636 n) (let loop ((n (* 2 n)) (m 1)) (cond ((< n 2) m) ((even? n) (loop (/ n 2) (* m 2))) (else (loop (/ (- n 1) 2) (A000040 m))))))

a(n) = A127301(A071163(n)) = A209637(A054429(n)).

A071162 Simple rewriting of binary expansion of n resulting A014486-codes for rooted binary trees with height equal to number of internal vertices. (Binary trees where at each internal vertex at least the other child is leaf).

Original entry on oeis.org

0, 2, 10, 12, 42, 44, 52, 56, 170, 172, 180, 184, 212, 216, 232, 240, 682, 684, 692, 696, 724, 728, 744, 752, 852, 856, 872, 880, 936, 944, 976, 992, 2730, 2732, 2740, 2744, 2772, 2776, 2792, 2800, 2900, 2904, 2920, 2928, 2984, 2992, 3024, 3040, 3412, 3416

Offset: 0

Antti Karttunen, May 14 2002

Essentially rewrites in binary expansion of n each 0 -> 01, 1X -> 1(rewrite X)0, where X is the maximal suffix after the 1-bit, which will be rewritten recursively (see the given Scheme-function). Because of this, the terms of the binary length 2n are counted by 2's powers, A000079.
In rooted plane (general) tree context, these are those totally balanced binary sequences (terms of A014486) where non-leaf subtrees can occur only as the rightmost branch (at any level of a general tree), but nowhere else. (Cf. A209642).
Also, these are exactly those rooted plane trees whose Łukasiewicz words happen to be valid asynchronous siteswap juggling patterns. (This was the original, albeit quite frivolous definition of this sequence for almost ten years 2002-2012. Cf. A071160.)

Antti Karttunen, Table of n, a(n) for n = 0..65535
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz

a(n) = A014486(A071163(n)) = A036044(A209642(n)) = A056539(A209642(n)).
A209859 provides an "inverse" function, i.e. A209859(a(n)) = n for all n.
Cf. A209636, A209637, A209862.

def a036044(n): return int(''.join('1' if i == '0' else '0' for i in bin(n)[2:][::-1]), 2)
def a209642(n):
    s=0
    i=1
    while n!=0:
        if n%2==0:
            n//=2
            s=4*s + 1
        else:
            n=(n - 1)//2
            s=(s + i)*2
        i*=4
    return s
def a(n): return 0 if n==0 else a036044(a209642(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, May 25 2017

(define (A071162 n) (let loop ((n n) (s 0) (i 1)) (cond ((zero? n) s) ((even? n) (loop (/ n 2) (+ s i) (* i 4))) (else (loop (/ (- n 1) 2) (* 2 (+ s i)) (* i 4))))))

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).

cp's OEIS Frontend

A209638 Sequence A209636 (or A209637) sorted into ascending order.

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A278542 a(n) = A046523(A209637(n)).

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A209636 Matula-numbers computed for rooted trees encoded by A071162/A071163.

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A071162 Simple rewriting of binary expansion of n resulting A014486-codes for rooted binary trees with height equal to number of internal vertices. (Binary trees where at each internal vertex at least the other child is leaf).

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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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