A075737 -id:A075737 - cp's OEIS frontend

A135958 a(n) = smallest prime p such that Fibonacci(p) has exactly n prime factors.

2, 3, 19, 37, 97, 229, 503, 367, 971

Offset: 0

Artur Jasinski, Dec 08 2007

A135959 gives the Fibonacci numbers. Cf. A000045, A050937, A075737, A090819, A135957.

a(n) = {p = 2; while (omega(fibonacci(p)) != n, p = nextprime(p+1)); p;} \\ Michel Marcus, Nov 08 2013

a(n) = prime(A135957(n)).

Edited and extended by David Wasserman, Mar 26 2008

A135952 Prime factors of composite Fibonacci numbers with prime indices (cf. A050937).

Original entry on oeis.org

37, 73, 113, 149, 157, 193, 269, 277, 313, 353, 389, 397, 457, 557, 613, 673, 677, 733, 757, 877, 953, 977, 997, 1069, 1093, 1153, 1213, 1237, 1453, 1657, 1753, 1873, 1877, 1933, 1949, 1993, 2017, 2137, 2221, 2237, 2309, 2333, 2417, 2473, 2557, 2593, 2749, 2777, 2789, 2797, 2857, 2909, 2917, 3217, 3253, 3313, 3517, 3557, 3733, 4013, 4057, 4177, 4273, 4349, 4357, 4513, 4637, 4733, 4909, 4933

Offset: 1

Artur Jasinski, Dec 08 2007

nonn

All numbers in this sequence are congruent to 1 mod 4. - Max Alekseyev.
If Fibonacci(n) is divisible by a prime p of the form 4k+3 then n is even. To prove this statement it is enough to show that (1+sqrt(5))/(1-sqrt(5)) is never a square modulo such p (which is a straightforward exercise).
The n-th prime p is an element of this sequence iff A001602(n) is prime and A051694(n)=A000045(A001602(n))>p. - Max Alekseyev

Hans Havermann, Table of n, a(n) for n = 1..5000

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852.

a = {}; k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], s = FactorInteger[Fibonacci[Prime[n]]]; c = Length[s]; Do[AppendTo[k, s[[m]][[1]]], {m, 1, c}]], {n, 2, 60}]; Union[k]

Edited, corrected and extended by Max Alekseyev, Dec 12 2007

A135959 a(n) = smallest Fibonacci number with prime index which has exactly n prime factors.

Original entry on oeis.org

1, 2, 4181, 24157817, 83621143489848422977, 322615043836854783580186309282650000354271239929

Offset: 0

Artur Jasinski, Dec 08 2007

nonn

a(6) has 105 digits.
a(7) = 22334640661774067356412331900038009953045351020683823507202893507476314037053.
Variant of A114722. [From R. J. Mathar, Oct 28 2008]

Cf. A000045, A050937, A075737, A134852, A135953-A135956, A135958.

a(n) = Fibonacci(A135958(n)).

Edited by David Wasserman, Mar 26 2008

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

A134790 Floor(prime Fibonacci(Prime(k))/Prime(k)).

Original entry on oeis.org

0, 1, 1, 8, 17, 93, 1245, 17732, 10081265, 63217341, 1195118711985005, 8140003186959622868813528, 139669360584622467747806013, 1324290912910022899017738237233285189213652972190967113265372469016533360

Offset: 1

Artur Jasinski, Nov 12 2007

nonn

			17732 is in the sequence because floor(514229/29) = 17732, where 29 is the 10th prime number and 514229 = Fibonacci(29) is also a prime. - _Bruno Berselli_, Jul 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..22

Cf. A050937, A090819, A134787, A134788, A134789.

Cf. A000040, A000045, A075737.

k = {}; Do[If[PrimeQ[Fibonacci[Prime[n]]], AppendTo[k, Floor[Fibonacci[Prime[n]]/Prime[n]]]], {n, 1, 100}]; k

A135968 Sum of the distinct prime factors of the Fibonacci number A050937(n).

Original entry on oeis.org

0, 150, 2974, 2443, 62158, 55946694, 2710261050, 555008010, 1547031, 46165377746, 95396546, 92180471494910, 1665088321801550, 364125780, 771601497990, 518283023, 8242065051309594, 32530503217194, 272602401466814027806, 5568053048227732238014, 85526725052226871

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

			a(2) = 150 = 37+113 because A050937(2) = 4181 = 37*113.
a(3) = 2974 = 557 + 2417 because A050937(3) = 1346269 = 557*2417.

Amiram Eldar, Table of n, a(n) for n = 1..203

Cf. A000045, A001605, A008472, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135954, A135955, A135956, A135957, A135958, A135959, A135969.

k = {}; Do[If[ ! PrimeQ[Fibonacci[Prime[n]]], b = FactorInteger[Fibonacci[Prime[n]]]; c =Length[FactorInteger[b]]; d = 0; Do[d = d + b[[r]][[1]], {r, 1, c}]; AppendTo[k, d]], {n, 1, 50}]; k

a(n) = A008472(A050937(n)). - R. J. Mathar, Dec 12 2007

Edited by R. J. Mathar, Dec 12 2007

a(19)-a(21) from Amiram Eldar, Oct 13 2024

A135969 Sum of the prime factors of A135953(n).

Original entry on oeis.org

150, 2974, 62158, 55946694, 2710261050, 555008010, 46165377746, 95396546, 92180471494910, 1665088321801550, 771601497990, 8242065051309594, 32530503217194, 272602401466814027806, 5568053048227732238014, 2811666624525811646469921614, 1966344318693363713266514

Offset: 1

Artur Jasinski, Dec 09 2007

nonn

A subsequence of A135968.

			a(1) = 150 = 37+113 because A135953(1) = 4181 = 37*113.
a(2) = 2974 = 557+2417 because A135953(2) = 1346269 = 557*2417.

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135954, A135955, A135956, A135957, A135958, A135959, A135968.

k = {}; Do[If[ ! PrimeQ[Fibonacci[Prime[n]]], b = FactorInteger[Fibonacci[Prime[n]]]; c = Length[FactorInteger[b]]; If[c == 2, d = b[[1]][[1]] + b[[2]][[1]]; AppendTo[k, d]]], {n, 1, 50}]; k
Total[Transpose[FactorInteger[#]][[1]]]&/@Select[Fibonacci[Prime[ Range[50]]], PrimeOmega[#]==2&] (* Harvey P. Dale, Feb 18 2012 *)

a(n) = A008472(A135953(n)). - R. J. Mathar, Nov 26 2008

Definition clarified by R. J. Mathar, Nov 26 2008

a(16)-a(17) from Amiram Eldar, Oct 23 2024

A277575 Primes p such that Fibonacci(prime(p)) is prime.

Original entry on oeis.org

2, 3, 5, 7, 23, 83, 97, 12161, 13681, 36467

Offset: 1

Bobby Jacobs, Oct 20 2016

Suggested by Alexander Adamchuk in A122534.

Primes in A119984.

Cf. A000040, A000045, A001605, A005478, A006450, A030426, A075737, A083668, A093308, A122534, A277284.

isok(p) = isprime(p) && isprime(fibonacci(prime(p))); \\ Michel Marcus, Oct 22 2016

a(n) = prime(A122534(n)).

a(n) = PrimePi(A277284(n)).

A117517 Numbers k such that F(2*k + 1) is prime where F(m) is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 14, 21, 23, 41, 65, 68, 179, 215, 216, 224, 254, 284, 285, 1485, 2361, 2693, 4655, 4838, 7215, 12780, 15378, 17999, 18755, 25416, 40919, 52455, 65010, 74045, 100553, 198689, 216890, 295020, 296844, 302355, 465758, 524948, 642803, 818003, 901529, 984360, 1452176

Offset: 1

Parthasarathy Nambi, Apr 26 2006

nonn

For F(k) to be prime, with k > 4, it is necessary but not sufficient for k to be prime. Hence after F(4) = 3, every prime F(m) is of the form F(2*k+1) for some k. Every prime divides some Fibonacci number. See also comment to A093062. - Jonathan Vos Post, Apr 29 2006

			If k=68 then F(2*k + 1) = 19134702400093278081449423917, a prime, so 68 is a term.

H. Dubner and W. Keller, New Fibonacci and Lucas Primes, Math. Comp. 68 (1999) 417-427.

Cf. A000045, A001605, A117595.

Cf. A001602, A022307, A030427, A051694, A075737, A083668, A099000.

[n: n in [0..1000] | IsPrime(Fibonacci(2*n+1))]; // Vincenzo Librandi, May 24 2016

Select[Range[0, 5000], PrimeQ[Fibonacci[2 # + 1]] &] (* Vincenzo Librandi, May 24 2016 *)

a(n) = (A083668(n)-1)/2. - R. J. Mathar, Jul 08 2009

a(n) = (A001605(n+1)-1)/2, n > 1. - Vincenzo Librandi, May 24 2016

More terms from Vincenzo Librandi, May 24 2016

A319230 Triple prime Fibonacci numbers: Fibonacci numbers f such that (1) f is prime, (2) f's index is prime, and (3) the sum of f's digits is prime.

Original entry on oeis.org

2, 5, 89, 514229, 433494437, 2971215073, 3061719992484545030554313848083717208111285432353738497131674799321571238149015933442805665949

Offset: 1

Harvey P. Dale, Sep 14 2018

a(8) = A000045(104911) has 21925 decimal digits. - Alois P. Heinz, Sep 17 2018

			514229 is prime; it is the 29th Fibonacci number and 29 is prime; and the sum of its digits is 23 and 23 is prime.

Cf. A000045, A001605, A005478, A007953, A075737, A319393. Subsequence of A111331.

Select[With[{nn=2000},Pick[Fibonacci[Range[nn]],PrimeQ[Range[nn]]]],PrimeQ[#] && PrimeQ[ Total[ IntegerDigits[#]]]&]

a(n) = A000045(A319393(n)).

cp's OEIS Frontend

A135958 a(n) = smallest prime p such that Fibonacci(p) has exactly n prime factors.

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A135952 Prime factors of composite Fibonacci numbers with prime indices (cf. A050937).

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A135959 a(n) = smallest Fibonacci number with prime index which has exactly n prime factors.

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

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A134790 Floor(prime Fibonacci(Prime(k))/Prime(k)).

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A135968 Sum of the distinct prime factors of the Fibonacci number A050937(n).

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A135969 Sum of the prime factors of A135953(n).

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A277575 Primes p such that Fibonacci(prime(p)) is prime.

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A117517 Numbers k such that F(2*k + 1) is prime where F(m) is a Fibonacci number.

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