A134860 -id:A134860 - cp's OEIS frontend

A095089 Fib101 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends as one, zero, one.

17, 59, 67, 101, 127, 211, 271, 313, 347, 373, 389, 449, 457, 491, 499, 593, 601, 643, 661, 677, 787, 821, 881, 983, 991, 1033, 1093, 1109, 1237, 1279, 1321, 1381, 1423, 1499, 1559, 1567, 1601, 1609, 1669, 1753, 1787, 1847, 1889, 1931, 1999

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A134860. Cf. A014417, A095069.

from sympy import fibonacci, primerange
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n): return str(a(n))[-3:]=="101"
print([n for n in primerange(1, 2001) if ok(n)]) # Indranil Ghosh, Jun 08 2017

A187951 Positions of 0 in A187950; complement of A187952.

Original entry on oeis.org

2, 4, 5, 7, 10, 12, 13, 15, 17, 18, 20, 23, 25, 26, 28, 31, 33, 34, 36, 38, 39, 41, 44, 46, 47, 49, 51, 52, 54, 57, 59, 60, 62, 65, 67, 68, 70, 72, 73, 75, 78, 80, 81, 83, 86, 88, 89, 91, 93, 94, 96, 99, 101, 102, 104, 106, 107, 109, 112, 114, 115, 117, 120, 122, 123, 125, 127, 128, 130, 133, 135, 136, 138, 140, 141, 143, 146, 148, 149, 151, 154

Offset: 1

Clark Kimberling, Mar 16 2011

nonn

See A187950.

Union of the Wythoff AAB-numbers A134860, the Wythoff BA-numbers A035336 and the Wythoff BB-numbers A101864. - Michel Dekking, Sep 14 2016

Cf. A187950, A187952.

Mathematica
```
(See A187950.)
```

A276757 Infinite Fibonacci word on the alphabet {1,2,3,4,5}.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 1, 2

Offset: 1

Michel Dekking, Sep 17 2016

nonn

Start with the infinite Fibonacci word A003849, which is 0100101001001010010... and replace each 0 by 1,2,3 and each 1 by 4,5.

The unique fixed point of the 4-block Fibonacci substitution 1 -> 12, 2 -> 3, 3 -> 45, 4 -> 12, 5 -> 3. Here the 4-blocks are coded as 0100 <-> 1, 1001 <-> 2, 0010 <-> 3, 0101 <-> 4, 1010 <-> 5.

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Michel Dekking and Michael Keane, On the conjugacy class of the Fibonacci dynamical system, arXiv preprint arXiv:1608.04487 [math.DS], 2016.

Cf. A000201, A001950, A003849, A134859, A035336, A003623, A134860, A101864, A270788, A138967.

Let A(n) = floor(n*phi), B(n) = n + floor(n*phi), i.e., A and B are the lower and upper Wythoff sequences, A = A000201, B = A001950. Then a(n) = 1 if n = A(A(A(k))) for some k; a(n) = 2 if n = B(A(k)) for some k; a(n) = 3 if n = A(B(k)) for some k; a(n) = 4 if n = A(A(B(k))) for some k; a(n) = 5 if n = B(B(k)) for some k.

A372302 Numbers k for which the Zeckendorf representation A014417(k) ends with "1001".

Original entry on oeis.org

6, 19, 27, 40, 53, 61, 74, 82, 95, 108, 116, 129, 142, 150, 163, 171, 184, 197, 205, 218, 226, 239, 252, 260, 273, 286, 294, 307, 315, 328, 341, 349, 362, 375, 383, 396, 404, 417, 430, 438, 451, 459, 472, 485, 493, 506, 519, 527, 540, 548, 561, 574, 582, 595, 603

Offset: 1

A.H.M. Smeets, Apr 25 2024

nonn

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A000045, A005614, A014417.
Tree of Zeckendorf subsequences of positive integers partitioned by their suffix part S (except initial term or offset in some cases). $ is the empty string. length(S) =
0        1        2        3        4        5        6       7
----------------------------------------------------------------------
$:       0:       00:      000:     0000:    00000:   000000:
A000027  A022342  A026274  A101345  A101642  notOEIS  notOEIS
                                                      100000: 0100000:
                                                      A035340  <------
                                             10000:
                                             A035339
                                    1000:    01000:
                                    A035338  <------
                  10:      010:     0010:
                  A035336  <------  A134861
                                    1010:    01010:
                                    A134863  <------
                           100:     0100:
                           A035337  <------
         1:       01:      001:     0001:
         A003622  <------  A134859  A151915
                                    1001:    01001:
                                    A372302  <------
                           101:     0101:
                           A134860  <------
Suffixes 10^n, where ^ means n times repeated concatenation, are the (n+1)-th columns in the Wythoff array A083412 and A035513 (n >= 0).

Equals {A134859}\{A151915}.
a(n) = A134863(n) - 1 = A035338(n) + 1.
a(n) = a(n-1) + 8 + 5*A005614(n-2) = a(n-1) + F(6) + F(5)*A005614(n-2), n > 1, where F(k) is the k-th Fibonacci number (A000045).

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A095089 Fib101 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends as one, zero, one.

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A187951 Positions of 0 in A187950; complement of A187952.

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Mathematica

A276757 Infinite Fibonacci word on the alphabet {1,2,3,4,5}.

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Formula

A372302 Numbers k for which the Zeckendorf representation A014417(k) ends with "1001".

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Author

Keywords

Links

Crossrefs

Formula