A030427 -id:A030427 - cp's OEIS frontend

A093062 a(n) = Fibonacci(prime(n)) - prime(Fibonacci(n)).

-1, 0, 2, 8, 78, 214, 1556, 4108, 28518, 513972, 1345808, 24156990, 165578670, 433491846, 2971210580, 53316283380, 956722012572, 2504730758802, 44945570173074, 308061521102198, 806515532933562, 14472334024479534, 99194853094422264, 1779979416004150202

Offset: 1

Dennis S. Kluk (mathemagician(AT)ameritech.net), May 08 2004

Composition of prime( ) and Fibonacci( ) is not commutative. Does a prime p ever divide Fibonacci(prime(p)) - prime(Fibonacci(p))?
Note that a(3) = 2 is the only prime element of the sequence. This is because after 2, all primes are odd; and the Fibonacci number F(n) is even only for n = 3k for some integer k [which relates to the fact that A082115 Fibonacci sequence (mod 3) is periodic with Pisano period 8]. Hence after a(1) = -1, Fibonacci(prime(n)) - prime(Fibonacci(n)) is always the difference of two odd numbers, hence is even. - Jonathan Vos Post, Jan 23 2006
Is a(i) ever divisible by i? Answer: yes. The quotient is an integer for i = 4, 28 and 30 through 63. - Dennis S. Kluk (mathemagician(AT)ameritech.net), Aug 16 2006

			a(11) = Fibonacci(prime(11)) - prime(Fibonacci(11)) = 1345808.

Chai Wah Wu, Table of n, a(n) for n = 1..84 (terms 1..41 from Harry J. Smith)

Cf. A000040, A000045, A030427.

Cf. A082115.

[Fibonacci(NthPrime(n)) - NthPrime(Fibonacci(n)): n in [1..30]]; // Vincenzo Librandi, Apr 10 2020

For[i=1, i<61, i++, Print[i, " ", Fibonacci[Prime[i]]-Prime[Fibonacci[i]]]]
Table[Fibonacci[Prime[n]]-Prime[Fibonacci[n]],{n,30}] (* Harvey P. Dale, Jul 02 2018 *)

a(n) = { fibonacci(prime(n)) - prime(fibonacci(n)) } \\ Harry J. Smith, Jun 20 2009

a(n) = Fibonacci(prime(n)) - prime(Fibonacci(n)).

A094894 a(n) = prime(Lucas(n)), Lucas numbers beginning at 2 (A000032).

Original entry on oeis.org

3, 2, 5, 7, 17, 31, 61, 109, 211, 383, 677, 1217, 2137, 3733, 6521, 11279, 19463, 33347, 56963, 97159, 165443, 280549, 474809, 801611, 1351841, 2273989, 3821689, 6412541, 10742339, 17974841, 30045019, 50163697, 83669419, 139420003, 232106309, 386086573, 641716373

Offset: 0

N. J. A. Sloane, Jun 15 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..82
Paul Cooijmans, Odds.

Cf. A027747, A030427, A000032, A000045, A000040.

[NthPrime(Lucas(n)): n in [0..40]]; // Vincenzo Librandi, Jan 14 2016

Table[ Prime[ Fibonacci[n + 1] + Fibonacci[n - 1]], {n, 0, 35}] (* Robert G. Wilson v, Jun 16 2004 *)
Table[Prime[LucasL[n]], {n, 0, 40}] (* Vincenzo Librandi, Jan 14 2016 *)

a(n) = prime(L(n)) = A000040(A000032(n)) = prime(Fib(n + 1) + Fib(n-1)), where Fib are the Fibonacci numbers(A000045) with Fib(-1) = -1.

More terms from Robert G. Wilson v, Jun 16 2004

a(34)-a(36) from Vincenzo Librandi, Jan 14 2016

A181058 a(n) = prime(Fibonacci(phi(n))), where prime = A000040, Fibonacci = A000045 and phi = A000010.

Original entry on oeis.org

2, 2, 2, 2, 5, 2, 19, 5, 19, 5, 257, 5, 827, 19, 73, 73, 7793, 19, 23159, 73, 827, 257, 196687, 73, 67931, 827, 23159, 827, 4528949, 73, 12717703, 7793, 67931, 7793, 563987, 827, 274253209, 23159, 563987, 7793, 2088145739, 827, 5738374519, 67931

Offset: 1

Carmine Suriano, Oct 01 2010

nonn

Phi is Euler's totient function A000010.

			a(7) = 19 since prime(fib(phi(7))) = prime(fib(6)) = prime(8) = 19 that is the 8th prime.

Amiram Eldar, Table of n, a(n) for n = 1..88

Cf. A000010, A000040, A000045, A030427, A065451, A181056.

f[n_] := Prime@ Fibonacci@ EulerPhi@ n; Array[f, 44] (* Robert G. Wilson v, Oct 02 2010 *)

A181058(n) = prime(fibonacci(eulerphi(n))); \\ Antti Karttunen, Dec 06 2017

a(n) = A000040(A065451(n)) = A030427(A000010(n)). - Antti Karttunen, Dec 06 2017

More terms from Robert G. Wilson v, Oct 02 2010

A343256 Difference between prime(Fibonacci(n+1)) and prime(Fibonacci(n)).

Original entry on oeis.org

0, 1, 2, 6, 8, 22, 32, 66, 118, 204, 366, 644, 1120, 1902, 3300, 5676, 9690, 16620, 28152, 47900, 80856, 136546, 230754, 387570, 651932, 1093174, 1832286, 3065822, 5122932, 8557788, 14272702, 23779968, 39592890, 65860910, 109471248, 181821502, 301795112

Offset: 1

Paolo Xausa, Apr 09 2021

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..83

Cf. A000040 (primes), A000045 (Fibonacci), A030427.

A343256[nmax_]:=Differences[Prime[Fibonacci[Range[nmax+1]]]];
A343256[20] (* Paolo Xausa, Jan 16 2023 *)

a(n) = prime(Fibonacci(n+1)) - prime(Fibonacci(n)).

a(n) = A030427(n+1) - A030427(n).

A093309 a(n) = prime(prime(Fibonacci(n))).

Original entry on oeis.org

3, 3, 5, 11, 31, 67, 179, 367, 797, 1621, 3259, 6353, 12301, 23209, 42979, 79559, 145547, 264091, 476981, 854353, 1523569, 2700559, 4765693, 8385679, 14683231, 25641599, 44620633, 77443423, 134053991, 231443561, 398799287, 685660127

Offset: 1

Cino Hilliard, Apr 25 2004

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..77 (terms 1..35 from Harry J. Smith)

Cf. A000040, A000045, A030427.

[NthPrime(NthPrime(Fibonacci(n))): n in [1..32]]; // Vincenzo Librandi, Dec 20 2015

Table[Prime[Prime[Fibonacci[n]]], {n, 1, 20}] (* G. C. Greubel, Dec 20 2015 *)

PARI
```
a(n) = prime(prime(fibonacci(n)))
```

a(n) = A000040(A030427(n)). - R. J. Mathar, Feb 06 2010

More terms from Harry J. Smith, Jun 20 2009

A113842 a(n) = Prime(tribonacci(n)).

Original entry on oeis.org

2, 2, 3, 7, 17, 41, 89, 193, 419, 859, 1759, 3607, 7247, 14551, 28793, 56893, 111863, 218839, 426583, 827851, 1603769, 3097121, 5966629, 11470489, 22004617, 42142883, 80570929, 153802489, 293176621, 558130051, 1061238359, 2015569583

Offset: 2

Jonathan Vos Post, Jan 23 2006

This is to the tribonacci sequence what A030427 is to the Fibonacci sequence.

			a(2) = prime(tribonacci(2)) = prime(1) = 2.
a(3) = prime(tribonacci(3)) = prime(1) = 2.
a(4) = prime(tribonacci(4)) = prime(2) = 3.
a(5) = prime(tribonacci(5)) = prime(4) = 7.
a(6) = prime(tribonacci(6)) = prime(7) = 17.

Chai Wah Wu, Table of n, a(n) for n = 2..68

Cf. A000040, A000073, A030427.

a(n) = A000040(A000073(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

A280105 a(n) = prime(Fibonacci(n)) written in base 2.

Original entry on oeis.org

10, 10, 11, 101, 1011, 10011, 101001, 1001001, 10001011, 100000001, 111001101, 1100111011, 10110111111, 101000011111, 1000110001101, 1111001110001, 11010010011101, 101101001110111, 1001101101100011, 10000100101011011, 11100010001110111, 110000000001001111

Offset: 1

Vincenzo Librandi, Dec 27 2016

nonn

Cf. A000040, A000045, A004685, A005478, A030427.

[Seqint(Intseq(NthPrime(Fibonacci(n)), 2)): n in [1..25]];

Table[FromDigits[IntegerDigits[Prime[Fibonacci[n]], 2]], {n, 1, 30}]

cp's OEIS Frontend

A093062 a(n) = Fibonacci(prime(n)) - prime(Fibonacci(n)).

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A094894 a(n) = prime(Lucas(n)), Lucas numbers beginning at 2 (A000032).

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A181058 a(n) = prime(Fibonacci(phi(n))), where prime = A000040, Fibonacci = A000045 and phi = A000010.

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A343256 Difference between prime(Fibonacci(n+1)) and prime(Fibonacci(n)).

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A093309 a(n) = prime(prime(Fibonacci(n))).

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A113842 a(n) = Prime(tribonacci(n)).

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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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