A014417 -id:A014417 - cp's OEIS frontend

A095080 Fibeven primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with zero.

2, 3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 47, 71, 73, 79, 83, 89, 97, 107, 109, 113, 131, 139, 149, 151, 157, 167, 173, 181, 191, 193, 199, 223, 227, 233, 241, 251, 257, 269, 277, 283, 293, 311, 317, 337, 353, 359, 367, 379, 397, 401, 409, 419, 421

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A022342. Union of A095082 and A095087. Cf. A095060, A095081.

F:= combinat[fibonacci]:
b:= proc(n) option remember; local j;
      if n=0 then 0
    else for j from 2 while F(j+1)<=n do od;
         b(n-F(j))+2^(j-2)
      fi
    end:
a:= proc(n) option remember; local p;
      p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
         if b(p)::even then break fi
      od; p
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 27 2016

F = Fibonacci;
b[n_] := b[n] = Module[{j},
     If[n == 0, 0, For[j = 2, F[j + 1] <= n, j++];
     b[n - F[j]] + 2^(j - 2)]];
a[n_] := a[n] = Module[{p},
     p = If[n == 1, 1, a[n - 1]]; While[True,
     p = NextPrime[p]; If[ EvenQ[b[p]], Break[]]]; p];
Array[a, 100] (* Jean-François Alcover, Jul 01 2021, after Alois P. Heinz *)

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))[-1]=="0"
print([n for n in primerange(1, 1001) if ok(n)]) # Indranil Ghosh, Jun 07 2017

A095081 Fibodd primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with one.

Original entry on oeis.org

17, 19, 43, 53, 59, 61, 67, 101, 103, 127, 137, 163, 179, 197, 211, 229, 239, 263, 271, 281, 307, 313, 331, 347, 349, 373, 383, 389, 433, 449, 457, 467, 491, 499, 509, 569, 577, 593, 601, 619, 643, 653, 661, 677, 739, 773, 787, 797, 821, 823

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A003622. Union of A095086 and A095089. Cf. A095061, A095080.

r = Map[Fibonacci, Range[2, 12]]; Select[Prime@ Range@ 144, Last@ Flatten@ Map[Position[r, #] &, Abs@ Differences@ NestWhileList[Function[k, k - SelectFirst[Reverse@ r, # < k &]], # + 1, # > 1 &]] == 1 &] (* Michael De Vlieger, Mar 27 2016, Version 10 *)

genit(maxx)={for(n=1,maxx,q=(n-1)+(n+sqrtint(5*n^2))\2; if(isprime(q), print1(q,",")));} \\ Bill McEachen, Mar 26 2016

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))[-1]=="1"
print([n for n in primerange(1, 1001) if ok(n)]) # Indranil Ghosh, Jun 07 2017

A095098 Fib001 numbers: those k for which the Zeckendorf expansion A014417(k) ends with two zeros and a final one.

Original entry on oeis.org

6, 9, 14, 19, 22, 27, 30, 35, 40, 43, 48, 53, 56, 61, 64, 69, 74, 77, 82, 85, 90, 95, 98, 103, 108, 111, 116, 119, 124, 129, 132, 137, 142, 145, 150, 153, 158, 163, 166, 171, 174, 179, 184, 187, 192, 197, 200, 205, 208, 213, 218, 221, 226, 229, 234, 239, 242

Offset: 1

Antti Karttunen, Jun 01 2004

The asymptotic density of this sequence is sqrt(5)-2. - Amiram Eldar, Mar 21 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A014417, A095086 (fib001 primes).

Set-wise difference of A003622 - A134860.

a[n_] = 2 Floor[(n + 1) GoldenRatio^2] - n - 3;
a /@ Range[100] (* Jean-François Alcover, Oct 28 2019, after Vladeta Jovovic *)

from sympy import fibonacci
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:]=="001"
print([n for n in range(1, 501) if ok(n)]) # Indranil Ghosh, Jun 08 2017

a(n) = 2*floor((n+1)*phi^2)-n-3, where phi = (1+sqrt(5))/2. - Vladeta Jovovic, Jul 05 2004

A331107 The sum of Zeckendorf-infinitary divisors of n = Product_{i} p(i)^r(i): divisors d = Product_{i} p(i)^s(i), such that the Zeckendorf expansion (A014417) of each s(i) contains only terms that are in the Zeckendorf expansion of r(i).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 27, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 108, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 99, 84, 144, 68

Offset: 1

Amiram Eldar, Jan 09 2020

First differs from A034448 at n = 16.

			a(16) = 27 since 16 = 2^4 and the Zeckendorf expansion of 4 is 101, i.e., its Zeckendorf representation is a set with 2 terms: {1, 3}. There are 4 possible exponents of 2: 0, 1, 3 and 4, corresponding to the subsets {}, {1}, {3} and {1, 3}. Thus 16 has 4 Zeckendorf-infinitary divisors: 2^0 = 1, 2^1 = 2, 2^3 = 8, and 2^4 = 16, and their sum is 1 + 2 + 8 + 16 = 27.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The number of Zeckendorf-infinitary divisors of n is in A318465.

Cf. A014417, A034448, A049417, A049418, A074847, A097863.

fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; Fibonacci[1 + Position[Reverse@fr, ?(# == 1 &)]]]; f[p, e_] := p^fb[e]; a[1] = 1; a[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); Array[a, 100] (* after Robert G. Wilson v at A014417 *)

Multiplicative with a(p^e) = Product_{i} (p^s(i) + 1), where s(i) are the terms in the Zeckendorf representation of e (A014417).

A095085 Fib000 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with three zeros.

Original entry on oeis.org

5, 13, 29, 47, 73, 89, 97, 107, 131, 149, 157, 173, 191, 199, 233, 241, 251, 293, 317, 419, 461, 479, 487, 521, 547, 563, 631, 673, 683, 691, 733, 751, 809, 827, 877, 911, 919, 937, 953, 971, 1013, 1021, 1039, 1063, 1097, 1123, 1249, 1259

Offset: 1

Antti Karttunen, Jun 01 2004

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

Intersection of A000040 and A101345.

Cf. A014417, A095065.

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:]=="000"
print([n for n in primerange(1, 1261) if ok(n)]) # Indranil Ghosh, Jun 08 2017

A095086 Fib001 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with two zeros and final 1.

Original entry on oeis.org

19, 43, 53, 61, 103, 137, 163, 179, 197, 229, 239, 263, 281, 307, 331, 349, 383, 433, 467, 509, 569, 577, 619, 653, 739, 773, 797, 823, 839, 857, 883, 907, 941, 967, 1009, 1051, 1061, 1069, 1103, 1129, 1153, 1171, 1187, 1213, 1229, 1289

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 A095098. Cf. A014417, A095066.

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:]=="001"
print([n for n in primerange(1, 1301) if ok(n)]) # Indranil Ghosh, Jun 08 2017

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

Original entry on oeis.org

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

A095353 Sum of 1-fibits in Zeckendorf-expansion A014417(p) summed for all primes p in range [Fib(n+1),Fib(n+2)[ (where Fib = A000045).

Original entry on oeis.org

0, 1, 1, 3, 2, 7, 7, 14, 23, 35, 56, 94, 155, 243, 402, 614, 1061, 1656, 2689, 4295, 6938, 11176, 18095, 29102, 46907, 75703, 122174, 197494, 317987, 514611, 829595, 1340861, 2166008, 3497040, 5645418, 9120129, 14733126, 23803219, 38460014

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratio a(n)/A095354(n) (i.e. average number of 1-fibits in Zeckendorf-expansions of primes p which Fib(n+1) <= p < Fib(n+2)) grows as: 1, 1, 1, 1.5, 2., 2.333333, 2.333333, 2.8, 3.285714, 3.181818, 3.5, 3.916667, 4.189189, 4.418182, 4.785714, 4.873016, 5.358586, 5.575758, 5.871179, 6.100852, 6.382705, 6.676225, 6.954266, 7.223132, 7.489542, 7.773978, 8.045173, 8.331323, 8.598659, 8.886546, 9.161734, 9.440489, 9.71936, 9.995484, 10.266207, 10.54327, 10.820602, 11.096084, 11.374267.

Ratio of that average compared to A010049(n)/A000045(n) (the expected value of that same sum computed for all integers in the same range) converges as: 1, 1, 0.666667, 0.9, 1, 1.037037, 0.919192, 0.99661, 1.063946, 0.945946, 0.96142, 1, 0.999059, 0.988519, 1.008389, 0.970278, 1.011305, 1.000122, 1.003368, 0.995592, 0.996635, 0.999338, 0.999601, 0.998575, 0.997298, 0.998427, 0.997837, 0.999078, 0.998056, 0.99941, 0.999296, 0.999567, 0.999834, 0.999811, 0.999265, 0.999347, 0.999451, 0.999382, 0.999555.

			a(1) = a(2) = 0, as there are no primes in ranges [1,2[ and [2,3[. a(3)=1 as in [3,5[ there is prime 3 with Fibonacci-representation 100. a(4)=3, as in [5,8[ there are primes 5 and 7, whose Fibonacci-representations are 1000 and 1010 respectively and we have three 1-fibits in total. a(5)=2, as in [8,13[ there is only one prime 11, with Zeckendorf-representation 10100.

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

Cf. A095336, A095298 (similar sums and ratios computed in binary system).

a(2) corrected by Chai Wah Wu, Jan 16 2020

A095731 Number of such primes p (A095730) such that Fib(n+1) <= p < Fib(n+2) (where Fib = A000045) and p's Zeckendorf-expansion A014417(p) is palindromic.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 3, 3, 0, 4, 8, 0, 15, 4, 0, 20, 42, 0, 44, 35, 0, 67, 147, 0, 231, 147, 0, 209, 538, 0, 833, 450, 0, 819, 2064, 0, 1701

Offset: 1

Antti Karttunen, Jun 12 2004

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

Cf. A095732, A095741.

A095082 Fib00 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with two zeros.

Original entry on oeis.org

3, 5, 11, 13, 29, 37, 47, 71, 73, 79, 89, 97, 107, 113, 131, 139, 149, 157, 173, 181, 191, 199, 223, 233, 241, 251, 257, 283, 293, 317, 359, 367, 401, 409, 419, 443, 461, 479, 487, 503, 521, 547, 563, 571, 587, 613, 631, 647, 673, 683, 691, 733

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

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

Cf. A095062. Intersection of A000040 and A026274. Union of A095085 and A095088.

list(lim)=my(v=List(), w=quadgen(20), phi=(1+w)/2, p2=phi^2, x=(2*phi-2)*p2, q);  lim=lim\1+1; while(xCharles R Greathouse IV, Nov 10 2021

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))[-2:]=="00"
print([n for n in primerange(1, 1001) if ok(n)]) # Indranil Ghosh, Jun 08 2017

cp's OEIS Frontend

A095080 Fibeven primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with zero.

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A095081 Fibodd primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with one.

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A095098 Fib001 numbers: those k for which the Zeckendorf expansion A014417(k) ends with two zeros and a final one.

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A331107 The sum of Zeckendorf-infinitary divisors of n = Product_{i} p(i)^r(i): divisors d = Product_{i} p(i)^s(i), such that the Zeckendorf expansion (A014417) of each s(i) contains only terms that are in the Zeckendorf expansion of r(i).

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A095085 Fib000 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with three zeros.

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Python

A095086 Fib001 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with two zeros and final 1.

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Python

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

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A095353 Sum of 1-fibits in Zeckendorf-expansion A014417(p) summed for all primes p in range [Fib(n+1),Fib(n+2)[ (where Fib = A000045).

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A095731 Number of such primes p (A095730) such that Fib(n+1) <= p < Fib(n+2) (where Fib = A000045) and p's Zeckendorf-expansion A014417(p) is palindromic.

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