A093062 -id:A093062 - cp's OEIS frontend

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

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

A180367 a(n) = Lucas(prime(n+1)) - prime(Lucas(n)), for Lucas numbers beginning at 2.

Original entry on oeis.org

0, 2, 6, 22, 182, 490, 3510, 9240, 63868, 1149468, 3009672, 54017304, 370246314, 969319296, 6643832358, 119218840092, 2139295466336, 5600748260454, 100501350226466, 688846502491240, 1803423556642478, 32361122671978600, 221806434537503870, 3980154972736116440

Offset: 0

Jonathan Vos Post, Aug 31 2010

nonn

Commutator of Primes and Lucas numbers. Some subtlety in indexing -- should we start with 0th Lucas number is 2, and 0th prime is 1? As shown here, I use "first" to mean the initial value as shown in P(n) and L(n), even though their indexing differs. This is to A093062 Fibonacci(prime(i))-prime(Fibonacci(i)) as Fibonacci is to Lucas.

			a(0) = 0 because the 1st prime is 2, and the third Lucas number is A000032(2) = 3; while the 1st Lucas number is 2, and the 2nd prime is 3; with 3-3=0.
a(1) = 2 because the 2nd prime is 3, and A000032(3) = 4; while the 2nd Lucas number is 1, and the first2 prime is 2; with 4-2=2.
a(2) = 6 because the 3rd prime is 5, and the 6th Lucas number (counting "2" as first) is A000032(5) = 11; while the 3rd Lucas number is 3, and the 3rd prime is 5; with 11-5=6.
a(3) = 29 - 7 = 22. a(4) = 199 - 17 = 182.

Amiram Eldar, Table of n, a(n) for n = 0..82 (terms 0..50 from Harvey P. Dale)

Cf. A000032, A000040, A094894, A180363, A093062.

A000032 := proc(n) option remember; if n <= 1 then op(n+1,[2,1]) ; else procname(n-1)+procname(n-2) ; end if; end proc:
A094894 := proc(n) ithprime(A000032(n)) ; end proc:
A180363 := proc(n) A000032(ithprime(n)) ; end proc:
A180367 := proc(n) A180363(n+1)-A094894(n) ; end proc: seq(A180367(n),n=0..25) ; # R. J. Mathar, Sep 01 2010

Table[LucasL[Prime[n+1]]-Prime[LucasL[n]],{n,0,30}] (* Harvey P. Dale, Jan 01 2021 *)

a(n) = L(prime(n+1)) - prime(L(n)) = A000032(A000040(n+1)) - A000040(A000032(n)) = A180363(n+1) - A094894(n).

Some indices corrected, 3 values corrected, and formulas signs swapped by R. J. Mathar, Sep 01 2010

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions