A001606 -id:A001606 - cp's OEIS frontend

A274333 Indices of Lucas numbers having exactly one primitive prime factor.

0, 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 27, 28, 30, 31, 33, 36, 37, 38, 41, 47, 49, 53, 54, 56, 61, 62, 66, 68, 70, 71, 72, 76, 78, 79, 80, 86, 90, 91, 96, 110, 113, 117, 120, 121, 136, 140, 144, 164, 168, 172, 178, 202, 203

Offset: 1

Arkadiusz Wesolowski, Jun 22 2016

nonn

0 together with numbers n such that A086600(n) = 1, except if n = 3.

The only primes in this sequence are the prime numbers in A001606, which gives the indices of prime Lucas numbers.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..102

Cf. A000032, A001606, A058036, A086600.

lst:=[]; pr:=1; for n in [0..203] do pd:=PrimeDivisors(Lucas(n)); d:=1; t:=0; for c in [1..#pd] do f:=pd[c]; if Gcd(pr, f) eq 1 then t+:=1; else d:=d*f; end if; end for; if t eq 1 then Append(~lst, n); end if; pr:=pr*Truncate(Lucas(n)/d); end for; lst;

A106627 Product L(n)*L_4(n), where L(n) are Lucas numbers and L_4(n) are Lucas 4-step numbers.

Original entry on oeis.org

8, 1, 9, 28, 105, 286, 918, 2871, 8977, 27892, 87084, 271635, 847182, 2641991, 8240325, 25700488, 80156033, 249994997, 779700654, 2431777739, 7584375260

Offset: 0

Jonathan Vos Post, May 11 2005

a(n) is semiprime iff n is an element of A001606 (an index of a prime Lucas number) and an element of A104577 (an index of a prime Lucas 4-step number). The only known such are n = 2, 8, 16, 19, 71, (through 145858).

			a(0) = 8 because L(0) * L_4(0) = 2 * 4.
a(1) = 1 because L(1) * L_4(1) = 1 * 1.
a(2) = 9 because L(2) * L_4(2) = 3 * 3.
a(3) = 28 because L(3) * L_4(3) = 4 * 7.
a(4) = 105 because L(4) * L_4(4) = 7 * 15.
a(5) = 286 because L(5) * L_4(5) = 11 * 26.
a(6) = 918 because L(6) * L_4(6) = 18 * 51.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,5,9,3,-2,1,-1).

Cf. A000032, A000040, A001358, A001606, A073446, A073817, A104576.

a:=[8,1,9,28,105,286,918,2871];; for n in [9..30] do a[n]:=a[n-1] +4*a[n-2]+5*a[n-3]+9*a[n-4]+3*a[n-5]-2*a[n-6]+a[n-7]-a[n-8]; od; a; # G. C. Greubel, Feb 19 2019

I:=[8,1,9,28,105,286,918,2871]; [n le 8 select I[n] else Self(n-1)+4*Self(n-2)+5*Self(n-3)+9*Self(n-4)+3*Self(n-5)-2*Self(n-6) + Self(n-7)-Self(n-8): n in [1..30]]; // G. C. Greubel, Feb 19 2019

LinearRecurrence[{1,4,5,9,3,-2,1,-1}, {8,1,9,28,105,286,918,2871}, 40] (* G. C. Greubel, Feb 19 2019 *)

my(x='x+O('x^40)); Vec((8-7*x-24*x^2-25*x^3-36*x^4-9*x^5+4*x^6 -x^7)/(1-x-4*x^2-5*x^3-9*x^4-3*x^5+2*x^6-x^7+x^8)) \\ G. C. Greubel, Feb 19 2019

((8-7*x-24*x^2-25*x^3-36*x^4-9*x^5+4*x^6-x^7)/(1-x-4*x^2-5*x^3 -9*x^4-3*x^5+2*x^6-x^7+x^8)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 19 2019

a(n) = A000032(n) * A073817(n).
a(n) = +a(n-1) +4*a(n-2) +5*a(n-3) +9*a(n-4) +3*a(n-5) -2*a(n-6) +a(n-7) -a(n-8). - R. J. Mathar, Dec 22 2010
G.f.: (8 -7*x -24*x^2 -25*x^3 -36*x^4 -9*x^5 +4*x^6 -x^7) / (1 -x -4*x^2 -5*x^3 -9*x^4 -3*x^5 +2*x^6 -x^7 +x^8). - Colin Barker, Jun 17 2012

A118841 Numbers k such that ceiling(phi^k) is prime.

Original entry on oeis.org

1, 2, 3, 4, 8, 16

Offset: 1

Eric W. Weisstein, May 01 2006

nonn

a(7) if it exists is greater than 250000. - Mark Rodenkirch, Feb 22 2020

Eric Weisstein's World of Mathematics, Phi-Prime

Cf. A118842, A001606.

Select[Range[20],PrimeQ[Ceiling[GoldenRatio^#]]&] (* James C. McMahon, Sep 14 2024 *)

c(n) = 3*fibonacci(n-1) + fibonacci(n-2) + (n % 2); \\ A169986(n) for n >= 1
for(n=1,10^7,if(ispseudoprime(c(n)),print1(n,", ")))
\\ Joerg Arndt, Feb 22 2020

a(7)-a(8) from Charles R Greathouse IV, Jul 30 2011
a(9)-a(11) from Charles R Greathouse IV, Aug 01 2011
Incorrect a(7)-a(11) removed by Mark Rodenkirch, Feb 22 2020

A216576 Numbers n such that n-th Lucas number is prime, but cannot be written in the form a^2 + 10*b^2.

Original entry on oeis.org

0, 2, 4, 7, 8, 11, 16, 19, 31, 47, 71, 79, 503, 863, 4787, 5851, 8467, 10691, 12251, 13963, 36779, 44507, 56003, 81671, 94823, 148091, 202667, 574219, 616787, 637751, 692147

Offset: 1

V. Raman, Sep 08 2012

nonn

Blair Kelly, Factorizations of Fibonacci and Lucas numbers

Cf. A000032, A001606.

A228916 Indices of primes in sequence A108300.

Original entry on oeis.org

1, 3, 9, 15, 39, 225, 231, 363, 687, 1299, 1335, 1809, 2367, 12735

Offset: 1

Vaclav Kotesovec, Sep 08 2013

Conjecture: for n>1 a(n) is a multiple of 3.
The indices greater than 363 yield probable primes. - Vaclav Kotesovec, Oct 28 2013
Next term > 100000. - Tyler Busby, Mar 29 2024

Cf. A108300, A209493, A001605, A001606, A096650, A099088.

seq=RecurrenceTable[{a[n]==3*a[n-1]+a[n-2],a[0]==1,a[1]==5},a,{n,1,1000}]; Select[Range[1000],PrimeQ[seq[[#]]]&]

A245472 Number of composite Lucas numbers between the prime Lucas numbers A005479(n) and A005479(n+1).

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 1, 2, 0, 1, 11, 5, 3, 5, 5, 7, 9, 7, 33, 199, 39, 149, 109, 3, 245, 233, 263, 3425, 5, 1057, 1889, 725, 2223, 1559, 1711, 485, 5019, 15979, 1329, 7727, 6661, 4833, 25667, 8177, 4973, 45233, 8033, 11429, 23567, 10111, 9465, 141625, 43139, 56175

Offset: 1

Michel Lagneau, Jul 23 2014

nonn

			a(1)=0 because A005479(1)= 2, A005479(2)= 3 and there are no composite Lucas number between 2 and 3 (the number 1 is not composite).
a(6)=2 because A005479(6)= 47, A005479(7)= 199 and there are 2 composite Lucas numbers between 47 and 199: 76 = 2^2*19 and 123 = 3*41.

Jens Kruse Andersen, Table of n, a(n) for n = 1..63

Cf. A000032, A001606, A245470.

with(combinat,fibonacci):with(numtheory):i:=0:a:=n->2*fibonacci(n-1)+fibonacci(n):for n from 3 to 100 do:if type (a(n), prime)=true then printf(`%d, `, i):i:=0:else i:=i+1:fi:od:

a(n) = A001606(n+1) - A001606(n) - 1, for n > 1.

A129745 Numbers k such that Lucas(4k)/7 is prime.

Original entry on oeis.org

5, 17, 19, 41, 43, 71, 1511, 2339, 3469, 4787, 7211, 9781, 14431

Offset: 1

Alexander Adamchuk, May 14 2007, May 16 2007

L(m) = Lucas(m) = Fibonacci(m-1) + Fibonacci(m+1). 7 = L(4) divides L(4*(1+2m)) since L(4m) = L(4)*L(4*(m-1)) - L(4*(m-2)).
Integer k is in this sequence iff k is prime and 4k belongs to A085726. - Max Alekseyev, May 16 2010
a(14) > 60000. - Michael S. Branicky, Aug 01 2024

Cf. A000032, A001606 (indices of prime Lucas numbers).

Cf. A074304 (numbers k such that Lucas(2k)/3 is prime).

a=7; b=47; Do[ c=7b-a; a=b; b=c; If[ PrimeQ[c/7], Print[n] ], {n, 3, 100}]

a(7) - a(10) from Stefan Steinerberger, May 17 2007
a(11) from Max Alekseyev, Nov 25 2007
a(12) from Alexander Adamchuk, May 15 2010
a(13) from Michael S. Branicky, Aug 01 2024

A194086 Second-smallest prime factor of the n-th Lucas number (beginning with 2), if composite, or 1 otherwise.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 1, 19, 41, 1, 7, 1, 281, 11, 1, 1, 3, 1, 2161, 29, 43, 461, 47, 101, 90481, 19, 14503, 19489, 3, 1, 4481, 199, 67, 29, 7, 1, 29134601, 79, 1601, 1, 3, 144481, 263, 11, 4969, 1, 769, 599786069, 41, 919, 103, 1, 3, 199, 10745088481, 229

Offset: 0

Jonathan Vos Post, Aug 14 2011

			a(3) = 2 because the 3rd of the Lucas numbers (beginning at 2) is A000032(3) = 4 = 2^2, hence depending on whether one means by second-smallest prime factor (i.e., distinct or not, with multiplicity or not) a(3) would be either 2 or 1.
a(10) is unambiguously 41, because L(10) = 123 = 3 * 41, and 41 is the second-smallest prime factor, with no issues of multiplicity or distinctness.

Cf. A000032, A000040, A001606, A005479, A193615.

Table[f = FactorInteger[LucasL[n]]; If[Length[f] > 1, f[[2, 1]], If[Length[f] == 1 && f[[1, 2]] > 1, f[[1, 1]], 1]], {n, 0, 70}] (* T. D. Noe, Aug 15 2011 *)

A216537 Numbers n such that the n-th Lucas number is prime and can be written in the form a^2 + 7*b^2.

Original entry on oeis.org

0, 4, 5, 7, 16, 17, 19, 37, 53, 71, 113, 353, 503, 613, 1361, 4787, 8467, 10691, 14449, 19649, 51169, 56003, 81671, 94823, 159521, 183089, 193201, 344293, 532277, 616787, 637751, 692147

Offset: 1

V. Raman, Sep 08 2012

nonn

Blair Kelly, Factorizations of Fibonacci and Lucas numbers

Cf. A000032, A001606.

A216538 Numbers n such that the n-th Lucas number is prime, but cannot be written in the form a^2 + 7*b^2.

Original entry on oeis.org

2, 8, 11, 13, 31, 41, 47, 61, 79, 313, 617, 863, 1097, 4793, 5851, 7741, 12251, 13963, 35449, 36779, 44507, 89849, 140057, 148091, 202667, 387433, 443609, 574219, 631181, 651821

Offset: 1

V. Raman, Sep 08 2012

nonn

Blair Kelly, Factorizations of Fibonacci and Lucas numbers

Cf. A000032, A001606.

cp's OEIS Frontend

A274333 Indices of Lucas numbers having exactly one primitive prime factor.

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A106627 Product L(n)*L_4(n), where L(n) are Lucas numbers and L_4(n) are Lucas 4-step numbers.

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A118841 Numbers k such that ceiling(phi^k) is prime.

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A216576 Numbers n such that n-th Lucas number is prime, but cannot be written in the form a^2 + 10*b^2.

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A228916 Indices of primes in sequence A108300.

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A245472 Number of composite Lucas numbers between the prime Lucas numbers A005479(n) and A005479(n+1).

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A129745 Numbers k such that Lucas(4k)/7 is prime.

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A194086 Second-smallest prime factor of the n-th Lucas number (beginning with 2), if composite, or 1 otherwise.

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A216537 Numbers n such that the n-th Lucas number is prime and can be written in the form a^2 + 7*b^2.

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