A090819 -id:A090819 - cp's OEIS frontend

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

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

A038672 Primes p such that the p-th Fibonacci number is composite.

Original entry on oeis.org

19, 31, 37, 41, 53, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 107, 109, 113, 127, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337

Offset: 1

Jeff Burch

nonn

Hans Havermann, Table of n, a(n) for n = 1..10000
Hans Havermann, Composite Prime-Index Fibonacci Numbers Factored

Cf. A090819 (essentially identical), A083668, A135952.

Rest[Select[Prime[Range[68]], ! PrimeQ[Fibonacci[#]] &]] (* Jayanta Basu, Jul 10 2013 *)
s={A083668}; Complement[Prime[Range[2, 10034]], s] (* Hans Havermann, Feb 12 2014 *)

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

A134791 a(n) = floor(log(Fibonacci(prime(k))/prime(k))), where k = A119984(n).

Original entry on oeis.org

-1, 0, 0, 2, 2, 4, 7, 9, 16, 17, 34, 57, 60, 166, 200, 201, 209, 237, 266, 267, 1420, 2263, 2582, 4470, 4646, 6933, 12289, 14789, 17311, 18039, 24449, 39369, 50472, 62555, 71250, 96762, 191209, 208726, 283920, 285676, 290979, 448242, 505208, 618634, 787250, 867638

Offset: 1

Artur Jasinski, Nov 12 2007

sign

Amiram Eldar, Table of n, a(n) for n = 1..55

Cf. A000045, A050937, A090819, A119984, A134787, A134788, A134789, A134790, A134792.

k = {}; Do[If[PrimeQ[Fibonacci[Prime[n]]], AppendTo[k, Floor[Log[Fibonacci[Prime[n]]/Prime[n]]]]], {n, 1, 200}]; k

a(21)-a(46) from Amiram Eldar, Oct 13 2024

A134792 a(n) = round(log(Fibonacci(prime(k))/prime(k))), where k = A119984(n).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 7, 10, 16, 18, 35, 57, 60, 166, 201, 201, 209, 238, 267, 268, 1421, 2263, 2583, 4471, 4647, 6934, 12289, 14789, 17312, 18039, 24450, 39370, 50472, 62555, 71250, 96762, 191210, 208727, 283921, 285676, 290980, 448242, 505208, 618634, 787251, 867638

Offset: 1

Artur Jasinski, Nov 12 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..55

Cf. A000045, A050937, A090819, A119984, A134787, A134788, A134789, A134790, A134791.

k = {}; Do[If[PrimeQ[Fibonacci[Prime[n]]], AppendTo[k, Round[Log[Fibonacci[Prime[n]]/Prime[n]]]]], {n, 1, 200}]; k

a(21)-a(46) 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

A134850 Number of primes between A075737(n) and A075737(n+1), including one bound.

Original entry on oeis.org

2, 3, 18, 27, 200, 2870, 39492, 22980943, 120106923, 2602986018837012

Offset: 1

Artur Jasinski, Nov 13 2007

Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000045, A050937, A075737, A090819, A134787, A307499.

a = {}; k = {}; Do[If[PrimeQ[Fibonacci[Prime[n]]], AppendTo[k, Fibonacci[Prime[n]]]], {n, 1, 100}];Do[AppendTo[a, PrimePi[k[[n + 1]]] - PrimePi[k[[n]]]], {n, 1, 9}]; a

Edited by N. J. A. Sloane, Oct 07 2008

a(10) using Kim Walisch's primecount, from Amiram Eldar, May 14 2023

A261312 Numbers n such that the n-th Fibonacci number is not divisible by any prime Fibonacci number.

Original entry on oeis.org

1, 2, 19, 31, 37, 38, 41, 53, 59, 61, 62, 67, 71, 73, 74, 79, 82, 89, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 134, 139, 142, 146, 149, 151, 157, 158, 163, 167, 173, 178, 179, 181, 191, 193, 194, 197, 199, 202, 206, 211, 214, 218, 223, 226, 227, 229, 233, 239, 241, 251, 254, 257

Offset: 1

Charles R Greathouse IV, Aug 14 2015

nonn

n is in this sequence if and only if it is not divisible by any member of A001605.
Conjecture: this sequence has natural density around 0.229. (The sequence has a natural density unless Fibonacci primes are much denser than expected.)
There are 232 terms up to 10^3, 2293 terms up to 10^4, 22869 terms up to 10^5, and 228896 terms up to 10^6. - Charles R Greathouse IV, Dec 11 2015

			The 19th Fibonacci is 4181 = 37*113, but neither 37 nor 113 is a Fibonacci number so 19 is in this sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000045, A001605, A090819.

is(n)=if(gcd(1155,n)>1||n%4==0, return(0)); my(f=factor(n)[,1]); for(i=1,#f, if(ispseudoprime(fibonacci(f[i])), return(0))); 1

A135960 Indices where records occur in A134852.

Original entry on oeis.org

1, 2, 4, 14, 37, 59, 144, 173

Offset: 1

Artur Jasinski, Dec 08 2007

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

a(7)-a(8) from Amiram Eldar, Sep 01 2019 (calculated from the b-file at A134852)

cp's OEIS Frontend

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

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A038672 Primes p such that the p-th Fibonacci number is composite.

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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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A134791 a(n) = floor(log(Fibonacci(prime(k))/prime(k))), where k = A119984(n).

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A134792 a(n) = round(log(Fibonacci(prime(k))/prime(k))), where k = A119984(n).

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

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A134850 Number of primes between A075737(n) and A075737(n+1), including one bound.

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A261312 Numbers n such that the n-th Fibonacci number is not divisible by any prime Fibonacci number.

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