A153867 -id:A153867 - cp's OEIS frontend

A005479 Prime Lucas numbers (cf. A000032).

2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149, 412670427844921037470771, 258899611203303418721656157249445530046830073044201152332257717521

Offset: 1

N. J. A. Sloane

It appears that a(n) is the intersection ( or a subset of the intersection ) of A113192[n], Primes that are the difference of two Lucas numbers and A113188[n], Primes that are the difference of two Fibonacci numbers, excluding A113192[1] = A113188[1] = 2. - Alexander Adamchuk, Aug 06 2006
For n>2 also: Primes which are the sum of four consecutive Fibonacci numbers, a(n) = A153867(n-2), cf. link to SeqFan list (Apr. 2014). - M. F. Hasler, Apr 24 2014
Conjectures: 7, 47 and 2207 are the only a(n) mod 10 = 7. They are also the only a(n) values where the Lucas index is not a prime.  See A001606 for the Lucas index values of these primes. See A266587 for the divisibility of Lucas numbers by powers of primes.  - Richard R. Forberg, Mar 24 2016

R. K. Guy, Unsolved Problems in Number Theory, Section A3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..28
J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260.
Harvey P. Dale and others, A005479 and A153867, SeqFan list, Apr 24 2014.
Blair Kelly, Factorizations of Lucas numbers
Ron Knott, The First 200 Lucas numbers and their factors.
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, Lucas Number

Cf. A000032, A001606, A113192, A113188.

Select[LucasL[Range[0,250]], PrimeQ] (* Harvey P. Dale, Nov 02 2011 *)

One further term (from the Knott web site) from Parthasarathy Nambi, Jun 27 2008

A153887 Smallest of five consecutive Fibonacci numbers whose sum is a prime number.

Original entry on oeis.org

0, 1, 2, 8, 89, 144, 2584, 121393, 196418, 832040, 20365011074, 4052739537881, 8944394323791464, 99194853094755497, 2880067194370816120, 14028366653498915298923761, 96151855463018422468774568

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,...

			For n=3,4, the Fibonacci indices of a(3)=2, a(4)=8, are 3,6 respectively. So a(3) + A000032(7)= 31,  a(4) + A000032(10) = 131. - _Vladimir Shevelev_, Apr 24 2014

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

a=0;b=1;c=1;d=2;lst={};Do[e=Fibonacci[n];p=a+b+c+d+e;If[PrimeQ[p],AppendTo[lst,a]];a=b;b=c;c=d;d=e,{n,4,6!}];lst
Select[Partition[Fibonacci[Range[1000]],5,1],PrimeQ[Total[#]]&][[All,1]] (* Harvey P. Dale, Dec 01 2016 *)

For n>=3, a(n) + A000032(m+4) is prime, where m is the Fibonacci index of a(n) (see example). This follows from the identity F(m+1) + F(m+2) + F(m+3) + F(m+4) = A000032(m+4), m>=0. - Vladimir Shevelev, Apr 24 2014

A153888 Second-to-smallest of five consecutive Fibonacci numbers such that sum of five consecutive Fibonacci numbers is prime number.

Original entry on oeis.org

1, 2, 3, 13, 144, 233, 4181, 196418, 317811, 1346269, 32951280099, 6557470319842, 14472334024676221, 160500643816367088, 4660046610375530309, 22698374052006863956975682, 155576970220531065681649693

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,...

Harvey P. Dale, Table of n, a(n) for n = 1..29

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

a=0;b=1;c=1;d=2;lst={};Do[e=Fibonacci[n];p=a+b+c+d+e;If[PrimeQ[p],AppendTo[lst,b]];a=b;b=c;c=d;d=e,{n,4,6!}];lst
Select[Partition[Fibonacci[Range[0,150]],5,1],PrimeQ[Total[#]]&][[All,2]] (* Harvey P. Dale, Dec 11 2018 *)

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 *)

A153891 Largest of five consecutive Fibonacci numbers such that the sum of the five consecutive Fibonacci numbers is prime.

Original entry on oeis.org

3, 8, 13, 55, 610, 987, 17711, 832040, 1346269, 5702887, 139583862445, 27777890035288, 61305790721611591, 679891637638612258, 19740274219868223167, 96151855463018422468774568, 659034621587630041982498215, 97605290770725966021179803308812675106786783237939047196728424115618

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

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, A153890.

a=0;b=1;c=1;d=2;lst={};Do[e=Fibonacci[n];p=a+b+c+d+e;If[PrimeQ[p],AppendTo[lst,e]];a=b;b=c;c=d;d=e,{n,4,6!}];lst
Transpose[Select[Partition[Fibonacci[Range[0,400]],5,1],PrimeQ[ Total[ #]]&]][[5]] (* Harvey P. Dale, Nov 14 2011 *)

One more term (a(17)) from Harvey P. Dale, Nov 14 2011

a(18) from Alois P. Heinz, Aug 31 2025

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A005479 Prime Lucas numbers (cf. A000032).

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A153887 Smallest of five consecutive Fibonacci numbers whose sum is a prime number.

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A153888 Second-to-smallest of five consecutive Fibonacci numbers such that sum of five consecutive Fibonacci numbers is prime number.

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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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A153891 Largest of five consecutive Fibonacci numbers such that the sum of the five consecutive Fibonacci numbers is prime.

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