A050937 -id:A050937 - cp's OEIS frontend

A134852 Number of distinct prime factors of the Fibonacci numbers in A050937.

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

Offset: 1

Artur Jasinski, Nov 13 2007

nonn

Charles R Greathouse IV and Amiram Eldar, Table of n, a(n) for n = 1..202 (terms 1..185 from Charles R Greathouse IV)

Cf. A000045, A001221, A001605, A050937, A075737, A090819, A134787, A134851.

a = {}; k = {}; Do[If[ ! PrimeQ[Fibonacci[Prime[n]]], c = Length[FactorInteger[Fibonacci[Prime[n]]]]; AppendTo[k, c]], {n, 1, 50}]; k

forprime(p=2,99,t=omega(fibonacci(p)); if(t!=1,print1(t", "))) \\ Charles R Greathouse IV, Feb 03 2014

a(n) = A001221(A050937(n)). - R. J. Mathar, May 03 2008

Edited by R. J. Mathar, May 03 2008

a(38)-a(87) from Charles R Greathouse IV, Feb 03 2014

A135956 Members of A050937 (nonprime Fibonacci numbers with prime index) with 5 or more distinct prime factors.

Original entry on oeis.org

322615043836854783580186309282650000354271239929, 1476475227036382503281437027911536541406625644706194668152438732346449273, 22334640661774067356412331900038009953045351020683823507202893507476314037053

Offset: 1

Artur Jasinski, Dec 08 2007

Conjecture: all numbers in this sequence are product of 5 or more sum of two squares

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135954, A135955, A135957.

k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], c = Length[FactorInteger[Fibonacci[Prime[n]]]]; Print[n]; If[c > 4, Print[Fibonacci[Prime[n]]]; AppendTo[k, Fibonacci[Prime[n]]]]], {n, 1, 100}]; k

A050937 INTERSECT { A051270 UNION A074969 UNION ... } = A050937 MINUS {A135955 UNION A135954 UNION A135953}. - R. J. Mathar, Jun 09 2008

Edited by R. J. Mathar, Jun 09 2008

A135954 Nonprime Fibonacci numbers with prime indices (A050937) that have exactly 3 prime factors.

Original entry on oeis.org

24157817, 44945570212853, 1500520536206896083277, 50095301248058391139327916261, 11463113765491467695340528626429782121, 30010821454963453907530667147829489881, 2211236406303914545699412969744873993387956988653, 103881042195729914708510518382775401680142036775841

Offset: 1

Artur Jasinski, Dec 08 2007

nonn

Conjecture: All numbers in this sequence are products of three sums of two squares.

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135955, A135956, A135957.

k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], c = Length[FactorInteger[Fibonacci[Prime[n]]]]; If[c == 3, AppendTo[k, Fibonacci[Prime[n]]]]], {n, 1, 50}]; k

f(n) = forprime(x=2, n, p=fibonacci(x); if(!isprime(p) && omega(p) == 3, print1(p", "))) \\ Georg Fischer, Feb 15 2025

a(6)-a(8) from Georg Fischer, Feb 15 2025

A135955 (Nonprime Fibonacci numbers with prime indices, A050937) which have exactly 4 prime factors.

Original entry on oeis.org

83621143489848422977, 6161314747715278029583501626149, 289450641941273985495088042104137, 5193981023518027157495786850488117, 66233869353085486281758142155705206899077

Offset: 1

Artur Jasinski, Dec 08 2007

nonn

Conjecture: All numbers in this sequence are products of four sums of two squares.

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135954, A135956, A135957.

k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], c = Length[FactorInteger[Fibonacci[Prime[n]]]]; If[c == 4, AppendTo[k, Fibonacci[Prime[n]]]]], {n, 1, 50}]; k

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

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

A134787 Numbers k such that Fibonacci(prime(k)) is not prime.

Original entry on oeis.org

1, 8, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 85, 86

Offset: 1

Artur Jasinski, Nov 12 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000720 (pi), A050937, A090819.

[n: n in [1..100]| not IsPrime(Fibonacci(NthPrime(n)))]; // Vincenzo Librandi, Jan 18 2013

k = {}; Do[If[ ! PrimeQ[Fibonacci[Prime[n]]], AppendTo[k, n]], {n, 1, 100}]; k
Select[Range[90],!PrimeQ[Fibonacci[Prime[#]]]&] (* Harvey P. Dale, Oct 16 2016 *)

is(n)=!isprime(fibonacci(prime(n))) \\ Charles R Greathouse IV, Feb 03 2014

a(n) = pi(A090819(n)). - Amiram Eldar, Oct 25 2024

Definition corrected by N. J. A. Sloane, Nov 12 2007

A090206 Nonprime Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 8, 21, 34, 55, 144, 377, 610, 987, 2584, 4181, 6765, 10946, 17711, 46368, 75025, 121393, 196418, 317811, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169

Offset: 1

Felix Tubiana, Jan 22 2004

It is possible to find a run of at least length n (not necessarily exactly n) such that n consecutive terms in this sequence are also consecutive in the sequence of Fibonacci numbers. However, it is not possible for such a run to be of exactly length n if n is even. - Alonso del Arte, Nov 23 2010

Some terms of this sequence have prime indices in the sequence of Fibonacci numbers (A000045), see A050937. - Alonso del Arte, Aug 16 2013

			34 is in the sequence as it is a Fibonacci number and it is composite, the product of 2 and 17.
55 is in the sequence as it is a Fibonacci number and it is composite, the product of 5 and 11.
89 is not in the sequence because, although it is a Fibonacci number, it is prime.

Cf. A000045, A005478, A050937, A182602.

Select[Fibonacci[Range[0, 50]], !PrimeQ[#] &] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)

A134851 Number of primes between A001605(n) and A001605(n+1).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 4, 1, 8, 9, 1, 39, 11, 1, 3, 10, 7, 1, 324, 208, 73, 442, 42, 498, 1122, 502, 508, 147, 1235, 2796, 2014, 2145, 1520, 4388, 15584, 2814, 11888, 274, 826, 24119, 8554, 16877, 24680, 11591, 11503, 63625, 22803, 6374, 92008, 115147, 79772, 157711, 3110

Offset: 1

Artur Jasinski, Nov 13 2007

nonn

Cf. A000045, A000720, A001605, A050937, A075737, A090819, A134787.

a = {}; k = {}; Do[If[PrimeQ[Fibonacci[n]], AppendTo[k, n]], {n, 1, 1000}]; Do[AppendTo[a, PrimePi[k[[n + 1]]] - PrimePi[k[[n]]]], {n, 1, 20}]; a

a(n) = primepi(A001605(n+1)) - primepi(A001605(n)). - Amiram Eldar, Sep 01 2019

More terms from Amiram Eldar, Sep 01 2019 and Sep 15 2024

A135953 (Nonprime Fibonacci numbers with prime indices) that have exactly 2 prime factors.

Original entry on oeis.org

4181, 1346269, 165580141, 53316291173, 956722026041, 2504730781961, 308061521170129, 806515533049393, 14472334024676221, 1779979416004714189, 573147844013817084101, 10284720757613717413913, 26925748508234281076009

Offset: 1

Artur Jasinski, Dec 08 2007

nonn

Conjecture: All numbers in this sequence are products of two sums of two squares, e.g. 4181 = 37*113 = (1^2+6^2)*(7^2+8^2), 1346269 = 557*2417 = (14^2+19^2)*(4^2+49^2).

Amiram Eldar, Table of n, a(n) for n = 1..42

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135954, A135955, A135956, A135957.

k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], c = Length[FactorInteger[Fibonacci[Prime[n]]]]; If[c == 2, AppendTo[k, Fibonacci[Prime[n]]]]], {n, 1, 50}]; k
Select[Fibonacci[Prime[Range[30]]],PrimeOmega[#]==2&] (* Harvey P. Dale, Feb 18 2012 *)

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A134852 Number of distinct prime factors of the Fibonacci numbers in A050937.

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A135956 Members of A050937 (nonprime Fibonacci numbers with prime index) with 5 or more distinct prime factors.

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A135954 Nonprime Fibonacci numbers with prime indices (A050937) that have exactly 3 prime factors.

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A135955 (Nonprime Fibonacci numbers with prime indices, A050937) which have exactly 4 prime factors.

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

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

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A134787 Numbers k such that Fibonacci(prime(k)) is not prime.

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A090206 Nonprime Fibonacci numbers.

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