A095085 -id:A095085 - cp's OEIS frontend

A095065 Number of fib000 primes (A095085) in range ]2^n,2^(n+1)].

0, 1, 1, 1, 1, 4, 9, 6, 19, 28, 54, 109, 210, 373, 707, 1316, 2497, 4827, 9127, 17467, 33212, 63161, 121404, 232455, 446846, 860466, 1658020, 3200462, 6184814, 11971998, 23184215, 44934259, 87179855, 169330402, 329113635

Offset: 1

Antti Karttunen, Jun 01 2004

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A095062, A095066, A095067, A095068, A095085.

a(n) = A095062(n) - A095068(n).

a(34)-a(35) from Amiram Eldar, Jun 13 2024

A101345 a(n) = Knuth's Fibonacci (or circle) product "2 o n".

Original entry on oeis.org

5, 8, 13, 18, 21, 26, 29, 34, 39, 42, 47, 52, 55, 60, 63, 68, 73, 76, 81, 84, 89, 94, 97, 102, 107, 110, 115, 118, 123, 128, 131, 136, 141, 144, 149, 152, 157, 162, 165, 170, 173, 178, 183, 186, 191, 196, 199, 204, 207, 212, 217, 220, 225, 228, 233, 238, 241, 246

Offset: 1

N. J. A. Sloane, Jan 26 2005

nonn

Numbers whose Zeckendorf representation ends with 000. - Benoit Cloitre, Jan 11 2014

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
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
Donald E. Knuth, Fibonacci multiplication, Appl. Math. Lett., Vol. 1, No. 1 (1988), pp. 57-60.

Second row of array in A101330.
Cf. A000201, A014417, A101642, A095085.
Set-wise difference of A026274 - A035337.

zeck[n_Integer] := 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-- ]; FromDigits[fr]]; kfp[n_, m_] := Block[{y = Reverse[ IntegerDigits[ zeck[ n]]], z = Reverse[ IntegerDigits[ zeck[ m]]]}, Sum[ y[[i]] * z[[j]] * Fibonacci[i + j + 2], {i, Length[y]}, {j, Length[z]}]];
Table[kfp[2, n], {n, 60}] (* Robert G. Wilson v, Feb 04 2005 *)
With[{r = Map[Fibonacci, Range[2, 14]]}, Rest[-1 + Position[#, Integer][[All, 1]]] &@ Table[1/1000 * Total@ Map[FromDigits@ PadRight[{1}, Flatten@ #] &@ Reverse@ Position[r, #] &, Abs@ Differences@ NestWhileList[Function[k, k - SelectFirst[Reverse@ r, # < k &]], n + 1, # > 1 &]], {n, 0, 250}]] (* _Michael De Vlieger, Jun 08 2017 *)
Array[2*Floor[(#+1)*GoldenRatio]+#-2 &, 100] (* Paolo Xausa, Mar 20 2024 *)

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 1 if str(a(n))[-3:]=="000" else 0 # Indranil Ghosh, Jun 08 2017

from math import isqrt
def A101345(n): return (n+1+isqrt(5*(n+1)**2)&-2)+n-2 # Chai Wah Wu, Aug 29 2022

a(n) = floor(phi^3*(n+1)) - 3 - floor(2*phi*(n+1)) + 2*floor(phi*(n+1)) where phi = (1+sqrt(5))/2. - Benoit Cloitre, Jan 11 2014

a(n) = 2*A000201(n+1) + n - 2. See the comments in A101642. - Michel Dekking, Dec 23 2019

More terms from David Applegate, Jan 26 2005

More terms from Robert G. Wilson v, Feb 04 2005

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

A095065 Number of fib000 primes (A095085) in range ]2^n,2^(n+1)].

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A101345 a(n) = Knuth's Fibonacci (or circle) product "2 o n".

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

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Python