A001605 -id:A001605 - cp's OEIS frontend

A105758 Indices of prime hexanacci (or Fibonacci 6-step) numbers A001592 (using offset -4).

3, 36, 37, 92, 660, 6091, 8415, 11467, 13686, 38831, 49828, 97148

Offset: 1

T. D. Noe, Apr 22 2005

No other n < 30000.
This sequence uses the convention of the Noe and Post reference. Their indexing scheme differs by 4 from the indices in A001592. Sequence A249635 lists the indices of the same primes (A105759) using the indexing scheme as defined in A001592. - Robert Price, Nov 02 2014 [Edited by M. F. Hasler, Apr 22 2018]
a(13) > 3*10^5. - Robert Price, Nov 02 2014

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A105759 (prime Fibonacci 6-step numbers), A249635 (= a(n) + 4), A001592.
Cf. A000045, A000073, A000078 (and A001631), A001591, A122189 (or A066178),  A079262, A104144,  A122265, A168082, A168083 (Fibonacci, tribonacci, tetranacci numbers and other generalizations).
Cf. A005478, A092836, A104535, A105757, A105761, ... (primes in these sequence).
Cf. A001605, A303263, A303264 (and A104534 and A247027), A248757 (and A105756), ... (indices of primes in A000045, A000073, A000078, ...).

a={1, 0, 0, 0, 0, 0}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, n]], {n, 30000}]; lst

a(n) = A249635(n) - 4. A105759(n) = A001592(A249635(n)) = A001592(a(n) + 4). - M. F. Hasler, Apr 22 2018

a(10)-a(12) from Robert Price, Nov 02 2014

Edited by M. F. Hasler, Apr 22 2018

A117595 Numbers n such that F(2*n - 1) is prime, where F(m) is a Fibonacci number.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 12, 15, 22, 24, 42, 66, 69, 180, 216, 217, 225, 255, 285, 286, 1486, 2362, 2694, 4656, 4839, 7216, 12781, 15379, 18000, 18756, 25417, 40920, 52456, 65011, 74046, 100554, 198690, 216891, 295021, 296845, 302356

Offset: 1

Parthasarathy Nambi, Apr 05 2006

nonn

See A001605, which is the main entry for Fibonacci primes, for the latest information. - N. J. A. Sloane, Jul 09 2016

Or, A001519(n) is prime. - Zak Seidov, Jul 04 2016

			If n=69 then F(2*n - 1) is a prime with 29 digits.

C. Caldwell's FibonacciPrime pages.
H. Dubner and W. Keller, New Fibonacci and Lucas Primes, Math. Comp. 68 (1999) 417-427.
PRP Top Records, Search for: F(n)

Cf. A000045, A001605 (Fibonacci(n) is prime), A001519.

Select[Range[2500], PrimeQ[Fibonacci[2# - 1]] &] (* Stefan Steinerberger, Apr 06 2006 *)

{ for(n=1,10000, if ( isprime( fibonacci(2*n-1) ), print1(n,","); ); ); } \\ R. J. Mathar, Apr 07 2006

2*a(n)-1 = A001605(n+1) for all odd A001605(n+1). - R. J. Mathar, Apr 07 2006

More terms from Stefan Steinerberger, T. D. Noe and R. J. Mathar, Apr 07 2006

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

A140987 Number of groups of order F(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 2, 2, 2, 1, 197, 1, 1, 6, 2, 1, 41, 1, 7, 4, 1, 1

Offset: 1

Jonathan Vos Post, Jul 28 2008

			a(6) = 5 because Fibonacci number F(6) = 8 and there are 5 groups of order 8.

GAP package GrpConst

Cf. A000001, A000040, A000045, A001358, A001605, A072381.

a(n) = A000001(A000045(n)).

Entries through a(17) checked by N. J. A. Sloane, Aug 01 2008

a(18)-a(23) from Eric M. Schmidt, Jun 19 2014

A153889 Middle of five consecutive Fibonacci numbers such that sum of five consecutive Fibonacci numbers is prime number.

Original entry on oeis.org

1, 3, 5, 21, 233, 377, 6765, 317811, 514229, 2178309, 53316291173, 10610209857723, 23416728348467685, 259695496911122585, 7540113804746346429, 36726740705505779255899443, 251728825683549488150424261, 37281903592600898879479448409585328515842582885579275203077366912825

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

0+1+1+2=3=7, 1+2+3+5+8=19, 2+3+5+8=13=31, 8+13+21+34+55=131, 89+144+233+377+610=1453, 144+233+377+610+987=2351,...

Cf. A000045, A001906, A000071, A001605, A153862, A153863, A153865, A153866, A153867, A153887, A153888.

a=0;b=1;c=1;d=2;lst={};Do[e=Fibonacci[n];p=a+b+c+d+e;If[PrimeQ[p],AppendTo[lst,c]];a=b;b=c;c=d;d=e,{n,4,6!}];lst

A153890 Second-to-largest of five consecutive Fibonacci numbers such that sum of five consecutive Fibonacci numbers is prime number.

Original entry on oeis.org

2, 5, 8, 34, 377, 610, 10946, 514229, 832040, 3524578, 86267571272, 17167680177565, 37889062373143906, 420196140727489673, 12200160415121876738, 59425114757512643212875125

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

0+1+1+2=3=7, 1+2+3+5+8=19, 2+3+5+8=13=31, 8+13+21+34+55=131, 89+144+233+377+610=1453, 144+233+377+610+987=2351,...

Cf. A000045, A001906, A000071, A001605, A153862, A153863, A153865, A153866, A153867, A153887, A153888, A153889

a=0;b=1;c=1;d=2;lst={};Do[e=Fibonacci[n];p=a+b+c+d+e;If[PrimeQ[p],AppendTo[lst,d]];a=b;b=c;c=d;d=e,{n,4,6!}];lst

A153892 Primes that are the sum of five consecutive Fibonacci numbers.

Original entry on oeis.org

7, 19, 31, 131, 1453, 2351, 42187, 1981891, 3206767, 13584083, 332484016063, 66165989928299, 146028309791690867, 1619478772188347101, 47020662244482792763, 229030451631542624193448579, 1569798068858809572115420691

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

Primes of the form F(k+3)+L(k+2), where F(k) and L(k) are the k-th Fibonacci number and Lucas number, respectively. This formula also gives that 3,2 and 5 are primes of the form F(k+3)+L(k+2), with k=-2, k=-1, k=0, respectively. - Rigoberto Florez, Jul 31 2022
Are there infinitely many primes of the form F(k+3)+L(k+2)? There are 47 primes of this form for k <= 80000. There are no such primes for 64000 <= k <= 80000. - Rigoberto Florez, Feb 26 2023
a(29) has 948 digits; a(30) has 1253 digits. - Harvey P. Dale, Jan 13 2013

			a(1) =  7 = 0+1+1+2+3 is prime;
a(2) = 19 = 1+2+3+5+8 is prime;
a(3) = 31 = 2+3+5+8+13 is prime, etc.

Harvey P. Dale, Table of n, a(n) for n = 1..29
Hsin-Yun Ching, Rigoberto Flórez, F. Luca, Antara Mukherjee, and J. C. Saunders, Primes and composites in the determinant Hosoya triangle, arXiv:2211.10788 [math.NT], 2022.
Hsin-Yun Ching, Rigoberto Flórez, F. Luca, Antara Mukherjee, and J. C. Saunders, Primes and composites in the determinant Hosoya triangle, The Fibonacci Quarterly, 60.5 (2022), 56-110.

Cf. A000045, A001906, A000071, A001605, A013655, A153862, A153863, A153865, A153866, A153867, A153887, A153888, A153889, A153890, A153891.

Select[Total/@Partition[Fibonacci[Range[0,150]],5,1],PrimeQ] (* Harvey P. Dale, Jan 13 2013 *)

A159977 a(n) = (smallest prime >= Fibonacci(n)) - Fibonacci(n).

Original entry on oeis.org

1, 1, 0, 0, 0, 3, 0, 2, 3, 4, 0, 5, 0, 2, 3, 4, 0, 7, 20, 14, 3, 2, 0, 13, 4, 10, 11, 16, 0, 23, 4, 4, 25, 10, 14, 35, 6, 24, 3, 2, 6, 7, 0, 20, 9, 48, 0, 5, 28, 18, 23, 14, 14, 11, 16, 10, 21, 4, 62, 13, 38, 12, 7, 16, 12, 19, 36, 28, 143, 32, 58, 29, 96, 100, 33, 2, 30, 27, 12, 62, 25, 46, 0

Offset: 1

Enoch Haga, Apr 28 2009

			a(1) = a(2) = 1 because Fibonacci(1) = Fibonacci(2) = 1, the smallest prime >= 1 is 2, and 2 - 1 = 1.
a(3) = a(4) = a(5) = 0 because Fibonacci(3)=2, Fibonacci(4)=3, and Fibonacci(5)=5 are all prime.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000045, A001605, A159978.

a:= n-> (f-> nextprime(f-1)-f)(combinat[fibonacci](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 04 2018

Table[If[PrimeQ[n],0,NextPrime[n]-n],{n,Fibonacci[Range[90]]}] (* Harvey P. Dale, Jul 22 2016 *)

F=1;G=0;for(i=1,100,print1(nextprime(F)-F,",");T=F;F+=G;G=T) \\ Hagen von Eitzen, Jul 20 2009

10 'FiboA 20 A=1:print A; 30 B=1:print B; 40 C=A+B:print C;:T=T+1 41 if C<>prmdiv(C) then print "<";nxtprm(C)-C;">":else print "<";0;">"; 50 D=B+C:print D; 51 if D<>prmdiv(D) then print "<";nxtprm(D)-D;">":else print "<";0;">"; 60 A=C:B=D:if T>22 then stop:else 40

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

a(n) = 0 <=> n in { A001605 }. - Alois P. Heinz, Feb 04 2018

More terms (cf. b-file) from Hagen von Eitzen, Jul 20 2009

Edited by Jon E. Schoenfield, Feb 04 2018

A177461 The smallest k such that Fibonacci(n)+k and Fibonacci(n)-k are both prime.

Original entry on oeis.org

0, 0, 0, 3, 0, 2, 3, 12, 0, 5, 0, 24, 3, 4, 0, 33, 48, 28, 57, 192, 0, 31, 12, 60, 81, 28, 0, 177, 108, 50, 345, 150, 168, 35, 6, 618, 735, 76, 18, 147, 0, 134, 111, 126, 0, 85, 642, 1146, 225, 92, 480, 219, 348, 466, 345, 72, 300, 89, 90, 312, 2025, 664, 168, 945, 276, 128

Offset: 3

Vladimir Joseph Stephan Orlovsky, May 09 2010

nonn

Indices where a(n)= 0 are provided by A001605.

			3 +- 0 -> primes, 5 +- 0 -> primes, 8 +- 3 -> primes, 13 +- 0 -> primes, 21 +- 2 -> primes, ...

Robert Israel, Table of n, a(n) for n = 3..1251

Cf. A000045, A001605, A047160.

A047160 := proc(n) for k from 0 to n-1 do if isprime(n-k) and isprime(n+k) then return k; end if; end do: return -1 ; end proc:
A177461 := proc(n) A047160(combinat[fibonacci](n)) ; end proc: # R. J. Mathar, Jan 23 2011

f[n_] := Block[{k}, If[n==2||OddQ[n], k=0, k=1]; While[!PrimeQ[n-k] || !PrimeQ[n+k], k+=2]; k]; Table[f[Fibonacci[n]], {n,3,100}]

a(n) = A047160(A000045(n)). - R. J. Mathar, Jan 23 2011

A263880 Safe primes 2p + 1 such that p is a Fibonacci prime.

Original entry on oeis.org

5, 7, 11, 179, 467, 21195998530602981465199287343010006825031720870818843865120019360285948694390966280586508792391539752259819

Offset: 1

Jonathan Sondow, Nov 02 2015

nonn

Same as safe primes q whose Sophie Germain prime (2q - 1)/2 is a Fibonacci number.
No other terms up to 2*Fibonacci(2904353) + 1, according to the list of indices of 49 Fibonacci (probable) primes in A001605.
In that range, the only safe Fibonacci prime is 5. Are there larger ones?
There are six primes 2p + 1 such that p is a Fibonacci prime, namely, a(1) through a(6). By contrast, in the same range there are only two primes 2p - 1 such that p is a Fibonacci prime, namely, 2p - 1 = 3 and 5, for p = 2 and 3. Is there some modular restriction to explain this bias in favor of 2p + 1 over 2p - 1 among Fibonacci primes p?

			179 is in the sequence because it is prime and (179 - 1)/2 = 89 = Fibonacci(11), which is also prime.

Wikipedia, Fibonacci prime
Wikipedia, Safe prime
Wikipedia, Sophie Germain prime

Cf. A000045, A005384, A005385, A005478, A155011.

2 * Select[Fibonacci[Range[2000]], And @@ PrimeQ[{#, 2 # + 1}] &] + 1

a(n) = 2*A155011(n) + 1.

cp's OEIS Frontend

A105758 Indices of prime hexanacci (or Fibonacci 6-step) numbers A001592 (using offset -4).

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A117595 Numbers n such that F(2*n - 1) is prime, where F(m) is a Fibonacci number.

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

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A140987 Number of groups of order F(n).

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A153889 Middle of five consecutive Fibonacci numbers such that sum of five consecutive Fibonacci numbers is prime number.

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A153890 Second-to-largest of five consecutive Fibonacci numbers such that sum of five consecutive Fibonacci numbers is prime number.

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A153892 Primes that are the sum of five consecutive Fibonacci numbers.

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A159977 a(n) = (smallest prime >= Fibonacci(n)) - Fibonacci(n).

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