A001606 -id:A001606 - cp's OEIS frontend

A113192 Primes that are the difference of two Lucas numbers; primes in A113191.

2, 3, 5, 7, 11, 17, 29, 43, 47, 73, 181, 197, 199, 293, 311, 503, 521, 839, 1361, 2131, 2203, 2207, 3571, 5749, 9349, 13763, 23633, 24469, 24473, 38239, 103483, 103681, 161983, 167759, 271367, 399601, 439081, 439157, 709283, 1692737, 3010349

Offset: 1

T. D. Noe, Oct 17 2005

nonn

The difference L(i)-L(j) equals the sum L(j+1)+...+L(i+2).

			The prime 181 is here because it is L(11)-L(6).

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

Cf. A000032 (Lucas numbers), A001606 (Lucas(n) is prime), A113193 (number of times that Lucas(n)-Lucas(i) is prime for i=0..n-3).

Lucas[n_] := Fibonacci[n+1]+Fibonacci[n-1]; lst={}; Do[p=Lucas[n]-Lucas[i]; If[PrimeQ[p], AppendTo[lst, p]], {n, 2, 40}, {i, 0, n-2}]; Union[lst]

A280104 a(n) = smallest prime factor of n-th Lucas number A000032(n), or 1 if there are none.

Original entry on oeis.org

2, 1, 3, 2, 7, 11, 2, 29, 47, 2, 3, 199, 2, 521, 3, 2, 2207, 3571, 2, 9349, 7, 2, 3, 139, 2, 11, 3, 2, 7, 59, 2, 3010349, 1087, 2, 3, 11, 2, 54018521, 3, 2, 47, 370248451, 2, 6709, 7, 2, 3, 6643838879, 2, 29, 3, 2, 7, 119218851371, 2, 11, 47, 2, 3, 709, 2

Offset: 0

Vincenzo Librandi, Dec 26 2016

nonn

From Robert Israel, Jan 05 2017: (Start)
If m and n are odd, m > 1 and m | n, then a(n) <= a(m).
a(n) = 2 if and only if 3 | n.
a(n) = 3 if and only if n is in A091999.
a(n) is never 5.
a(n) = 7 if and only if n is in A259755.
a(n) = A000032(n) if and only if n is in A001606.
(End)

Robert Israel, Table of n, a(n) for n = 0..1000
Fibonacci and Lucas Factorizations

Cf. A000032, A001606, A020639, A079451 (same for largest prime factor), A091999, A139044, A144293, A259755, A279623.

Column k=2 of A238899 (for n>=2).

[2,1] cat [Minimum(PrimeDivisors(Lucas(n))): n in [2..60]];

lucas:= n -> combinat:-fibonacci(n+1)+combinat:-fibonacci(n-1):
spf:= proc(n) local F;
  F:= remove(hastype,ifactors(n,easy)[2],symbol);
  if F <> [] then return min(seq(f[1],f=F)) fi;
min(numtheory:-factorsec(n))
end proc:
spf(1):= 1:
map(spf @ lucas, [$0..200]); # Robert Israel, Jan 05 2017

f[n_]:=(FactorInteger@LucasL@n)[[1, 1]]; Array[f, 60, 0]

a000032(n) = fibonacci(n+1)+fibonacci(n-1)
a(n) = if(a000032(n-1)==1, 1, factor(a000032(n-1))[1, 1]) \\ Felix Fröhlich, Dec 26 2016

a(n) = A020639(A000032(n)). - Felix Fröhlich, Dec 26 2016

Offset changed from Bruno Berselli, Dec 27 2016

A076697 Indices of record values in A079451, largest prime factor of Lucas numbers A000032.

Original entry on oeis.org

0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 26, 31, 37, 41, 47, 53, 61, 68, 71, 76, 79, 86, 113, 136, 164, 172, 178, 202, 218, 229, 262, 278, 284, 307, 313, 328, 353, 373, 436, 443, 458, 487, 503, 557, 577, 586, 613, 617, 746, 751, 758, 863, 914

Offset: 0

Shane Findley, Oct 25 2002

nonn

From  M. F. Hasler, Apr 09 2025: (Start)
Original name: Next-to-largest factor of Lucas(n).
The offset 0 is coherent with the fact that the initial term is a starting value rather than a record value.
When A000032(n) is prime (<=> n is in A001606), it necessarily sets a new record for the largest prime factor, since A000032 is increasing from the second term on. Therefore, A001606 is a subsequence. (End)

Ron Knott, Lucas numbers.
Douglas S. McNeil, in reply to Robert Israel, A076697, SeqFan google group, April 9, 2025.

Cf. A000042 (Lucas numbers, starting with 2), A079451 (largest prime factor of these).

Cf. A001606 (Indices of prime Lucas numbers: a subsequence).

A076697_first(n, m=0)=vector(n,i, i>1 || n=-1; until(mA079451(n++), m), );n) \\ M. F. Hasler, Apr 09 2025

def A076697(n):
    try: terms, M = A076697.terms, A076697.M
    except AttributeError: A076697.terms = terms = [0]; A076697.M = M = 2
    while len(terms) <= n: terms.append(next(i for i in range(terms[-1]+1, 1<<59)
        if M < (M:=max(A079451(i),M)))); A076697.M = M
    return terms[n] # M. F. Hasler, Apr 10 2025

New definition and data corrected and extended by M. F. Hasler, Apr 09 2025

A123677 Primes p such that Lucas(prime(p)) is prime, where Lucas = A000032.

Original entry on oeis.org

3, 5, 7, 11, 13, 71, 113, 643, 769, 13681, 51929

Offset: 1

Alexander Adamchuk, Oct 05 2006

These are the primes in A120561.

Numbers n such that Lucas(prime(n)) is prime are listed in A120561; indices of prime Lucas numbers are listed in A001606.

Cf. A000032, A123678, A120561, A001606, A119984, A122534, A277290.

a(n) = prime(A123678(n)).

a(n) = pi(A277290(n)). - Bobby Jacobs, Oct 30 2016

51929 found by Henri Lifchitz, from Jens Kruse Andersen, Jul 24 2014

A123678 Numbers n such that Lucas(prime(prime(n))) is prime, where Lucas(k) = A000032(k).

Original entry on oeis.org

2, 3, 4, 5, 6, 20, 30, 117, 136, 1616, 5313

Offset: 1

Alexander Adamchuk, Oct 05 2006

Indices of prime Lucas numbers are listed in A001606.
Numbers n such that Lucas(prime(n)) is prime are listed in A120561.
Primes in A120561 are listed in A123677(n) = prime(a(n)).

Cf. A000032, A123677, A120561, A001606, A277290.

Cf. A119984, A122534 (Numbers n such that Fibonacci(prime(prime(n))) is prime).

a(n) = pi(A123677(n)).

a(n) = pi(pi(A277290(n))). - Bobby Jacobs, Oct 30 2016

5313 found by Henri Lifchitz, from Jens Kruse Andersen, Jul 24 2014

A193495 Number of odd divisors of Lucas(n), Lucas numbers beginning at 2.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 8, 4, 4, 8, 4, 4, 6, 4, 12, 2, 4, 4, 8, 16, 8, 2, 4, 8, 8, 2, 24, 4, 16, 32, 8, 2, 8, 4, 16, 8, 8, 2, 20, 24, 4, 8, 8, 8, 32, 2, 4, 32, 4, 32, 24, 4, 4, 16, 16, 2, 8, 4, 8, 64, 4, 16, 12, 2, 4

Offset: 0

Jonathan Vos Post, Jul 28 2011

This is to A193292 as A000032 Lucas numbers (beginning at 2) is to A000045 Fibonacci numbers.

			a(18) = 8 because L(18) = 5778 = 2 * 3^3 * 107 whose odd divisors are 8 in number: {1, 3, 9, 27, 107, 321, 963, 2889}. a(n) = 2 iff n is in A001606 (Indices of prime Lucas numbers).

Amiram Eldar, Table of n, a(n) for n = 0..1000

Cf. A000032, A001227, A001606, A193292.

Table[luc = LucasL[n]; r = IntegerExponent[luc, 2]; DivisorSigma[0, luc/2^r], {n, 0, 100}] (* T. D. Noe, Jul 29 2011 *)
Table[Count[Divisors[LucasL[n]],?OddQ],{n,0,80}] (* _Harvey P. Dale, Mar 28 2019 *)

a(n) = A001227(A000032(n)).

A073446 Product L(n)S(n), where L(n) are Lucas numbers and S(n) are Lucas 3-step numbers = A000032(n) A001644(n).

Original entry on oeis.org

6, 1, 9, 28, 77, 231, 702, 2059, 6157, 18316, 54489, 162185, 482678, 1436397, 4274853, 12722028, 37861085, 112675763, 335326230, 997940307, 2969899037, 8838503884, 26303639349, 78280380217, 232964641030, 693309407681

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 01 2002

a(n) is also the trace of the matrix R^n, where R is the Kronecker product of the Fibonacci matrix (Fibomatrix): first row (1,1), second row (1,0), times the Tribomatrix: first row (1,1,0), second row (1,0,1), third row (1,0,0).

a(n) is semiprime iff n is an element of A001606 (an index of a prime Lucas number) and an element of A104576 (an index of a prime Lucas 3-step number). The only known such are n = 2, 4, 7, 8 (through 67661). - Jonathan Vos Post, May 10 2005

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Elia, Derived Sequences, The Tribonacci Recurrence and Cubic Forms, The Fibonacci Quarterly 39.2 (2001): 107-109.
F. T. Howard, A Tribonacci Identity, The Fibonacci Quarterly 39.4 (2001): 352-357.
Index entries for linear recurrences with constant coefficients, signature (1,4,5,2,-1,1).

Cf. A000032, A000040, A001358, A001606, A001644, A104576.

a:=[6,1,9,28,77,231];; for n in [7..40] do a[n]:=a[n-1]+4*a[n-2] +5*a[n-3]+2*a[n-4]-a[n-5]+a[n-6]; od; a; # G. C. Greubel, Feb 19 2019

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6) )); // G. C. Greubel, Feb 19 2019

CoefficientList[Series[(6-5x-16x^2-15x^3-4x^4+x^5)/(1-x-4x^2-5x^3-2x^4 +x^5-x^6), {x, 0, 50}], x]

my(x='x+O('x^40)); Vec((6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6)) \\ G. C. Greubel, Feb 19 2019

((6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 19 2019

a(n) = a(n-1)+4*a(n-2)+5*a(n-3)+2*a(n-4)-a(n-5)+a(n-6), a(0)=6, a(1)=1, a(2)=9, a(3)=28, a(4)=77, a(5)=231.

G.f.: (6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6).

A120561 Numbers n such that Lucas(prime(n)) is prime, where Lucas = A000032.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 11, 12, 13, 15, 16, 18, 20, 22, 30, 65, 71, 96, 112, 113, 150, 184, 218, 643, 645, 769, 982, 1059, 1304, 1464, 1649, 1695, 2208, 3776, 3899, 4626, 5236, 5684, 7988, 8700, 9143, 13013, 13681, 14641, 16590, 17433, 18198, 29529, 32870, 37234, 43994, 47150, 50373, 51420, 51929, 52953, 55965, 71398, 82258

Offset: 1

Alexander Adamchuk, Aug 07 2006, Oct 05 2006

nonn

All prime Lucas numbers A000032[n] have indices that are prime, zero or a power of 2. It is a conjecture that all indices of prime Lucas numbers are prime, except n = 0, 4, 8, 16.
Indices of prime Lucas numbers are listed in A001606[n] = {0,2,4,5,7,8,11,13,16,17,19,31,37,41,47,53,61,...}.
Primes in a(n) are listed in A123677[n] = {3,5,7,11,13,71,113,643,769,13681,...} Primes p such that Lucas[Prime[p]] is prime.
Numbers n such that Lucas[Prime[Prime[n]]] is prime are listed in A123678[n] = PrimePi[A123677[n]] = {2,3,4,5,6,20,30,117,136,1616,...}.

Cf. A000032, A119984. Cf. A001606 - Indices of prime Lucas numbers.

Cf. A123677, A123678.

Select[ Range[300], PrimeQ[ Fibonacci[ Prime[ # ] - 1 ] + Fibonacci[ Prime[ # ] + 1 ]] & ]

a(n) = PrimePi(A001606(n+4)) for n>5.

a(52)-a(60) (from A001606) from Jens Kruse Andersen, Jul 24 2014

A121533 Fibonacci-Lucas prime twins: Prime Fibonacci numbers Fibonacci(k) such that Lucas numbers Lucas(k) = Fibonacci(k-1) + Fibonacci(k+1) are also prime.

Original entry on oeis.org

3, 5, 13, 89, 233, 1597, 2971215073

Offset: 1

Alexander Adamchuk, Aug 05 2006

nonn

Indices for Fibonacci-Lucas prime twins are A080327(n) = {4, 5, 7, 11, 13, 17, 47, ...}. Corresponding Lucas-Fibonacci prime twins are A121534(n) = {7, 11, 29, 199, 521, 3571, 6643838879, ...}. Probable primes Fibonacci(148091) and Lucas(148091) are the next probable Fibonacci-Lucas and Lucas-Fibonacci prime twins. They have 30949 and 30950 digits.

			a(1) = 3 because Fibonacci(4) = 3 is prime and Lucas(4) = 5 is also prime.

Cf. A121534, A000045, A005478, A001605, A001606, A000032, A000204, A005479, A080327.

Do[f=Fibonacci[n]; l=Fibonacci[n-1]+Fibonacci[n+1]; If[PrimeQ[f]&&PrimeQ[l], Print[{f,l}]], {n,10000}]

A168033 Primes p such that floor(phi^p) is prime.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, 51169, 56003, 81671, 89849, 94823, 140057, 148091, 159521, 183089, 193201, 202667, 344293, 387433, 443609, 532277, 574219, 616787, 631181, 637751, 651821, 692147, 901657, 1051849

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 17 2009

nonn

Primes in A059791. - Charles R Greathouse IV, Jul 29 2011

Also primes in A001606. - Michel Marcus, Oct 21 2016

Cf. A001606, A059791.

[p: p in PrimesUpTo(2000)| IsPrime(Lucas(p))]; // Vincenzo Librandi, Jul 11 2019

$MaxExtraPrecision=6!; Select[Prime[Range[5! ]],PrimeQ[Floor[GoldenRatio^# ]]&]

phi=(1+sqrt(5))/2;forprime(p=2,1e3,if(isprime(floor(phi^p)),print1(p", "))) \\ Charles R Greathouse IV, Jul 29 2011

a(22)-a(32) from Charles R Greathouse IV, Jul 29 2011

More terms (using A001606) from Joerg Arndt, Jul 11 2019

cp's OEIS Frontend

A113192 Primes that are the difference of two Lucas numbers; primes in A113191.

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A280104 a(n) = smallest prime factor of n-th Lucas number A000032(n), or 1 if there are none.

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A076697 Indices of record values in A079451, largest prime factor of Lucas numbers A000032.

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A123677 Primes p such that Lucas(prime(p)) is prime, where Lucas = A000032.

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A123678 Numbers n such that Lucas(prime(prime(n))) is prime, where Lucas(k) = A000032(k).

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A193495 Number of odd divisors of Lucas(n), Lucas numbers beginning at 2.

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A073446 Product L(n)*S(n), where L(n) are Lucas numbers and S(n) are Lucas 3-step numbers = A000032(n) * A001644(n).

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A120561 Numbers n such that Lucas(prime(n)) is prime, where Lucas = A000032.

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A073446 Product L(n)S(n), where L(n) are Lucas numbers and S(n) are Lucas 3-step numbers = A000032(n) A001644(n).