A001605 -id:A001605 - cp's OEIS frontend

A080345 a(n) is the number of prime factors in Fibonacci(prime(n)); that is, in the Fibonacci number whose index is the n-th prime.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 1, 2, 4, 2, 3, 2, 2, 2, 2, 1, 1, 3, 4, 2, 4, 4, 2, 2, 3, 3, 2, 2, 4, 2, 4, 4, 2, 5, 3, 4, 3, 2, 3, 3, 4, 2, 2, 3, 4, 2, 4, 4, 4, 3, 2, 3, 5, 4, 2, 1, 7, 5, 4, 3, 3, 2, 2, 4, 3, 4, 1, 1, 5, 5, 1, 3, 5, 3, 2, 3, 4, 3, 4, 6, 1, 3, 4, 3

Offset: 1

T. D. Noe, Feb 16 2003

In all known examples, Fibonacci(prime(n)) is squarefree, in which case a(n) is well-defined, i.e., the number of distinct prime factors equals the total number of prime factors. But if for some n, Fibonacci(prime(n)) has a repeated prime factor, then a(n) is not well-defined. - Jonathan Sondow, Oct 22 2015

			a(12) = 3 because the 12th prime is 37 and Fibonacci(37) = 24157817 = 73 * 149 * 2221 has 3 prime factors. - clarified by _Jonathan Sondow_, Oct 21 2015

Amiram Eldar, Table of n, a(n) for n = 1..222 (terms 1..168 from T. D. Noe)
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A001605, A022307.

Table[Length[FactorInteger[Fibonacci[Prime[n]]]], {n, 60}]
PrimeNu[Fibonacci[Prime[Range[100]]]] (* Harvey P. Dale, Mar 13 2016 *)

a(n) = omega(fibonacci(prime(n))); \\ Michel Marcus, Oct 22 2015

a(n) = A001221(A000045(A000040(n))). - Michel Marcus, Oct 22 2015

A099000 Indices k such that the k-th prime is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 6, 24, 51, 251, 3121, 42613, 23023556, 143130479, 2602986161967491

Offset: 1

Rick L. Shepherd, Nov 06 2004

From Hugo Pfoertner, Jan 06 2020: (Start)
The computation of the next two terms, corresponding to the primes F(131) = A005478(13) = 1066340417491710595814572169, and F(137) = A005478(14) = 19134702400093278081449423917, should already be within reach with current (2020) technology, e.g. with Kim Walisch's "primecount" program, which allows massive parallelization. An exact determination of the following term a(15), which corresponds to F(359), is beyond any imaginable technical possibility.
Estimates for a(13)-a(15), found by using the PARI program from A121046 in a bisection loop, with an accuracy that corresponds to the shown number of digits, are as follows:
a(13) = primepi(F(131)) ~= 1.741898800848...*10^25,
a(14) = primepi(F(137)) ~= 2.9848914766265...*10^26,
a(15) = primepi(F(359)) ~= 2.78114064956041656819790214151422895...*10^72.
(End)

Cf. A001605 (n-th Fibonacci number is prime), A005478 (Prime Fibonacci numbers).

Cf. A121046.

PrimePi[Select[Fibonacci[Range[80]], PrimeQ]]

print1("1, 2");forprime(p=5,47,if(isprime(fibonacci(p)),print1(", "primepi(fibonacci(p))))) \\ Charles R Greathouse IV, Aug 21 2011

a(n) = A000720(A005478(n)). - M. F. Hasler, Aug 21 2011

a(11) from Ryan Propper, Oct 16 2005

a(12) from Charles R Greathouse IV, Aug 21 2011

A122534 Numbers k such that Fibonacci(prime(prime(k))) is prime.

Original entry on oeis.org

1, 2, 3, 4, 9, 23, 25, 1456, 1616, 3865

Offset: 1

Alexander Adamchuk, Sep 18 2006

The corresponding primes are {2,5,89,1597,99194853094755497,...}.
Numbers k such that A093308(k) is prime.
A277575(n) = prime(a(n)) is a prime in A119984.

Cf. A000040, A000045, A119984, A083668, A093308, A075737, A001605, A006450, A005478, A030426, A277284, A277575.

a(n) = PrimePi(A277575(n)) = PrimePi(PrimePi(A277284(n))). - Bobby Jacobs, Oct 26 2016

A140362 Semiprimes pq that divide the sum of the squares of their divisors, 1+p^2+q^2+(pq)^2.

Original entry on oeis.org

10, 65, 20737

Offset: 1

Mohamed Bouhamida, Jul 22 2008, Jul 27 2008

6 is the smallest integer n which is the product of two distinct primes and which divides the sum of the cubes of the divisors of n. Are there other numbers with this property?
Using Pell equations and a Fibonacci identity, Max Alekseyev and I have shown that all terms are the product of prime Fibonacci numbers whose indices are twin primes. The first three terms are Fib(3)*Fib(5), Fib(5)*Fib(7) and Fib(11)*Fib(13). The other two known terms are Fib(431)*Fib(433) and Fib(569)*Fib(571), huge numbers that are in the b-file. The sequence probably has no additional terms. - T. D. Noe, Jul 27 2008
Let a, b, c and d be consecutive odd-indexed Fibonacci numbers. Then it can be proved that 1 + b^2 + c^2 + (bc)^2 = abcd, which shows that bc divides 1 + b^2 + c^2 + (bc)^2. Hence if b and c are prime, then bc is in this sequence. - T. D. Noe, Jul 27 2008
Empirical search suggests that A067558(a(n))/A032741(a(n)) = a(n). A032741(a(n)) = 3 for all n by definition of semiprime. A067558(a(n)) must also then be divisible by 3. a(n) can be called the n-th "perfect mean square aliquot number". - William Krier, Dec 16 2024

			10 divides (1^2 + 2^2 + 5^2) giving 3 - the number of proper divisors of semiprime 10.
65 divides (1^2 + 5^2 + 13^2) giving 3 - the number of proper divisors of semiprime 65.
20737 divides (1^2 + 89^2 + 233^2) giving 3 - the number of proper divisors of semiprime 20737.

T. D. Noe, Table of n, a(n) for n=1..5
T. Cai, D. Chen, and Y. Zhang, Perfect numbers and Fibonacci primes, arXiv:1310.0898 [math.NT], 2013-2014.
T. Cai, D. Chen, and Y. Zhang, Perfect numbers and Fibonacci primes (II), arXiv:1406.5684 [math.NT], 2014 (see case m=1 in Table 1).

Cf. A000045, A001605, A046762.

isok(n) = sigma(n, 2) - n^2 == 3*n; \\ Michel Marcus, Jun 24 2014

A279795 Numbers n such that F(n) and F(n-2) are both prime where F(n) = A000045(n).

Original entry on oeis.org

5, 7, 13, 433, 571

Offset: 1

Altug Alkan, Dec 18 2016

a(6) > 2904353. - Daniel Suteu, Dec 23 2016
Terms n of A001605 such that n-2 is also a term of A001605. Surprisingly, the first 4 terms minus 2, { 3, 5, 11, 431 }, are the first four terms of A101315 which also relates to simultaneously prime { m+2, F(m) and F(m)+2 }, but where F is a different function, m -> (m-1)^2 + 1. - M. F. Hasler, Dec 24 2016
Larger primes of the Fibonacci prime pairs in A073340. - Bobby Jacobs, Jan 18 2017

			13 is a term because Fibonacci(13) = 233 and Fibonacci(11) = 89 are both prime.

Cf. A000045, A001605, A006512, A073340, A101315, A281087.

Select[Range[10^4], Times @@ Boole@ Map[PrimeQ@ Fibonacci@ # &, {#, # - 2}] > 0 &] (* Michael De Vlieger, Jan 21 2017 *)
Flatten[Position[Partition[Fibonacci[Range[580]],3,1],?(AllTrue[ {#[[1]],#[[3]]},PrimeQ]&),1,Heads->False]]+2 (* _Harvey P. Dale, Oct 01 2021 *)

isok(n) = isprime(fibonacci(n)) && isprime(fibonacci(n-2)); \\ Michel Marcus, Jan 14 2017

a(n) = A281087(n) + 2. - Bobby Jacobs, Jan 18 2017

A303263 Indices of primes in tribonacci sequence A000073.

Original entry on oeis.org

4, 6, 7, 11, 87, 98, 215, 802, 4202, 18699, 96879

Offset: 1

M. F. Hasler, Apr 18 2018

T = A000073 is defined by T(n+1) = T(n) + T(n-1) + T(n-2), T(2) = 1, T(1) = T(0) = 0.

The largest terms correspond to unproven probable primes T(a(n)).

Cf. A000073 (= T), A092836 (= T(a(n))), A092835 (= a(n) - 1).

Cf. A001605 (indices of primes in Fibonacci numbers A000045).

-1 + Position[LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 10^4], ?PrimeQ][[All, 1]] (* _Michael De Vlieger, Apr 21 2018 *)

a(n,N=4,S=vector(N,i,i>N-2))={for(i=N,oo,ispseudoprime(S[i%N+1]=2*S[(i-1)%N+1]-S[i%N+1])&&!n--&&return(i))}

a(n) = A092835(n) + 1 = index of A092836(n) in A000073.

A050180 Numbers m such that prime(m) + Fibonacci(m) is prime.

Original entry on oeis.org

1, 3, 12, 24, 42, 60, 132, 303, 312, 450, 564, 1236, 1419, 2952, 10098, 12060, 13848, 17229, 22356, 37896, 251904

Offset: 1

Jason Earls, Oct 06 2001

a(22) > 256737. - J.W.L. (Jan) Eerland, Jun 22 2022

Cf. A004397, A001605.

[n: n in [1..2000] | IsPrime(NthPrime(n)+Fibonacci(n))]; // Vincenzo Librandi, Jul 29 2016

Do[ If[ PrimeQ[ Prime[n] + Fibonacci[n]], Print[n] ], {n, 1, 150000} ]
Select[Range[10000], PrimeQ[Fibonacci[#] + Prime[#]] &] (* Vincenzo Librandi, Jul 29 2016 *)
DeleteCases[ParallelTable[If[PrimeQ[Prime[n]+Fibonacci[n]],n,a],{n,0,256737}],a] (* J.W.L. (Jan) Eerland, Jun 22 2022 *)

for(n=1,10^5, if(isprime(prime(n)+fibonacci(n)),print1(n," ")))

More terms from Robert G. Wilson v, Oct 08 2001
a(18)-a(19) from Vincenzo Librandi, Apr 10 2020
a(20) from J.W.L. (Jan) Eerland, Dec 23 2021
a(21) from J.W.L. (Jan) Eerland, Jun 22 2022

A068329 Arithmetic derivative of Fibonacci numbers > 0.

Original entry on oeis.org

0, 0, 1, 1, 1, 12, 1, 10, 19, 16, 1, 384, 1, 42, 437, 491, 1, 4164, 150, 4388, 6341, 288, 1, 155472, 30035, 754, 115271, 142474, 1, 1530588, 2974, 1084624, 1802069, 5168, 2555363, 46594656, 503939, 1406531

Offset: 1

Reinhard Zumkeller, Feb 27 2002

nonn

Iain Fox, Table of n, a(n) for n = 1..400 (first 200 terms from T. D. Noe)

Cf. A000045, A003415, A001605 (where a(n) = 1).

a:=Concatenation([0,0],List(List([3..40],n->Factors(Fibonacci(n))),i->Product(i)*Sum(i,j->1/j))); # Muniru A Asiru, Oct 31 2018

Ad:=func; [n le 2 select 0 else Ad(Fibonacci(n)): n in [1..40]]; // Bruno Berselli, Oct 22 2013

ad[1] = 0; ad[n_] := Module[{f = FactorInteger[n]}, Total[n*f[[All, 2]]/ f[[All, 1]]]]; a[n_] := ad[Fibonacci[n]]; Array[a, 40] (* Jean-François Alcover, Feb 22 2018 *)

a(n) = my(f = factor(n=fibonacci(n))~); sum(i=1, #f, n/f[1, i]*f[2, i]) \\ Iain Fox, Oct 29 2018

from sympy import fibonacci, factorint
def A068329(n):
    f = fibonacci(n)
    return sum((f*e//p for p,e in factorint(f).items())) if n > 2 else 0 # Chai Wah Wu, Jun 12 2022

a(n) = A003415(A000045(n)).

A073340 Fibonacci prime pairs: the indices of each pair differ by two and the relevant Fibonacci numbers are both prime.

Original entry on oeis.org

3, 5, 5, 7, 11, 13, 431, 433, 569, 571

Offset: 1

Harvey P. Dale, Aug 25 2002

There are no other Fibonacci prime pairs up to Fibonacci(104911). (See A001605.) Are there any larger terms?

			The 431st Fibonacci number and the 433rd Fibonacci number are both prime and their indices differ by 2.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Rev. ed. 1997, p. 46.

Cf. A000045, A001605, A279795, A281087.

Flatten[Select[Partition[Select[Range[3000], PrimeQ[Fibonacci[ # ]]&], 2, 1], #[[2]] - #[[1]] == 2 &]]

from sympy import isprime
def afind(limit):
  i, fnm2, fnm1 = 1, 1, 1
  while i < limit:
    if isprime(fnm2) and isprime(fnm2 + fnm1):
      print(i, i+2, sep=", ", end=", ")
    i, fnm2, fnm1 = i+1, fnm1, fnm2 + fnm1
afind(600) # Michael S. Branicky, Mar 05 2021

Offset changed to 1 by Joerg Arndt, Jan 18 2017

a(1) and a(2) prepended by Bobby Jacobs, Jan 18 2017

A082602 Number of primes between successive Fibonacci numbers (including possibly the Fibonacci numbers themselves).

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 3, 5, 8, 11, 17, 24, 37, 55, 85, 126, 198, 297, 458, 704, 1088, 1674, 2602, 4029, 6263, 9738, 15187, 23705, 36981, 57909, 90550, 142033, 222855, 349862, 549903, 865019, 1361581, 2145191, 3381318, 5334509, 8419528, 13298631, 21014892

Offset: 1

Hauke Worpel (hw1(AT)email.com), May 23 2003

nonn

			a(10) = 8 because the 10th Fibonacci number is 55, the 11th is 89 and the eight primes between them are 59, 61, 67, 71, 73, 79, 83 and 89.

Amiram Eldar, Table of n, a(n) for n = 1..122 (calculated using the b-file at A054782)

Cf. A000045, A054782, A076777.

Cf. A001605, A005478.

[#PrimesInInterval(Fibonacci(n-1), Fibonacci(n)): n in [2..45]]; // Vincenzo Librandi, Jul 13 2017

lst={};Do[p=0;Do[If[PrimeQ[a],p++ ],{a,Fibonacci[n],Fibonacci[n+1]}];AppendTo[lst,p],{n,50}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)

{ a(n)= c=0; forprime(N=fibonacci(n),fibonacci(n+1),c=c+1); return(c); }

Corrected and extended by Rick L. Shepherd, May 26 2003

a(43)-a(44) from Vincenzo Librandi, Jul 13 2017

cp's OEIS Frontend

A080345 a(n) is the number of prime factors in Fibonacci(prime(n)); that is, in the Fibonacci number whose index is the n-th prime.

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A099000 Indices k such that the k-th prime is a Fibonacci number.

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Mathematica

PARI

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Extensions

A122534 Numbers k such that Fibonacci(prime(prime(k))) is prime.

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A140362 Semiprimes pq that divide the sum of the squares of their divisors, 1+p^2+q^2+(pq)^2.

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PARI

A279795 Numbers n such that F(n) and F(n-2) are both prime where F(n) = A000045(n).

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Mathematica

PARI

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A303263 Indices of primes in tribonacci sequence A000073.

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Mathematica

PARI

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A050180 Numbers m such that prime(m) + Fibonacci(m) is prime.

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Magma

Mathematica

PARI

Extensions

A068329 Arithmetic derivative of Fibonacci numbers > 0.

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