A113188 -id:A113188 - 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

A007298 Sums of consecutive Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 16, 18, 19, 20, 21, 26, 29, 31, 32, 33, 34, 42, 47, 50, 52, 53, 54, 55, 68, 76, 81, 84, 86, 87, 88, 89, 110, 123, 131, 136, 139, 141, 142, 143, 144, 178, 199, 212, 220, 225, 228, 230, 231, 232, 233, 288, 322

Offset: 1

N. J. A. Sloane, Jan 02 2000

Also the differences between two Fibonacci numbers, because the difference F(i+2) - F(j+1) equals the sum F(j) + ... + F(i). - T. D. Noe, Oct 17 2005; corrected by Patrick Capelle, Mar 01 2008

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

Cf. A000045, A050939.
Cf. A113188 (primes that are the difference of two Fibonacci numbers).
Cf. A219114 (numbers whose squares are here).

isA007298 := proc(n)
    local i,Fi,j,Fj ;
    for i from 0 do
        Fi := combinat[fibonacci](i) ;
        for j from i do
            Fj :=combinat[fibonacci](j) ;
            if Fj-Fi = n then
                return true;
            elif Fj-Fi > n then
                break;
            end if;
        end do:
        Fj :=combinat[fibonacci](i+1) ;
        if Fj-Fi > n then
            return false;
        end if;
    end do:
end proc:
for n from 0 to 100 do
    if isA007298(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 25 2016

Union[Flatten[Table[Fibonacci[n]-Fibonacci[i], {n, 14}, {i, n}]]] (* T. D. Noe, Oct 17 2005 *)
isA007298[n_] := Module[{i, Fi, j, Fj}, For[i = 0, True, i++, Fi = Fibonacci[i]; For[j = i, True, j++, Fj = Fibonacci[j]; Which[Fj - Fi == n, Return@True, Fj - Fi > n, Break[]]]; Fj := Fibonacci[i + 1]; If[Fj - Fi > n, Return@False]]];
Select[Range[0, 1000], isA007298] (* Jean-François Alcover, Nov 16 2023, after R. J. Mathar *)

A130233(n)=log(sqrt(5)*n+1.5)\log((1+sqrt(5))/2)
list(lim)=my(v=List([0]),F=vector(A130233(lim),i,fibonacci(i)),s,t); for(i=1,#F, s=0; forstep(j=i,1,-1, s+=F[j]; if(s>lim, break); listput(v,s))); Set(v) \\ Charles R Greathouse IV, Oct 06 2016

log a(n) >> sqrt(n). - Charles R Greathouse IV, Oct 06 2016

A113239 Prime differences of tribonacci numbers.

Original entry on oeis.org

2, 3, 5, 11, 17, 23, 31, 37, 43, 79, 193, 491, 503, 653, 883, 1201, 10607, 19009, 19469, 19489, 34963, 35809, 46499, 223273, 223313, 391231, 409817, 410731, 532159, 634061, 754549, 1383769, 1389533, 2552621, 2555753, 3311233, 4477453, 4700621

Offset: 1

Jonathan Vos Post, Oct 19 2005

A113238 is the difference set of tribonacci numbers. A113188-A113194 deal with difference sets of Fibonacci numbers and Lucas numbers and primes in those difference sets.

			a(1) = 2 because 4 - 2 = 2 where 4 and 2 are tribonacci numbers.
a(2) = 3 because 7 - 4 = 3 where 7 and 4 are tribonacci numbers.
a(3) = 5 because 7 - 2 = 5 where 7 and 2 are tribonacci numbers.
a(4) = 11 because 13 - 2 = 11 where 13 and 2 are tribonacci numbers.
a(5) = 17 because 24 - 7 = 17 where 24 and 7 are tribonacci numbers.

Cf. A000040, A000073, A113188-A113194, A113238.

Select[Union[Flatten[Differences/@Subsets[Drop[LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 29],2],{2}]]],PrimeQ] (* James C. McMahon, Jun 23 2024 *)

Intersection of primes A000040 and difference set of tribonacci numbers A113238. Positive prime values of {A000073(i) - A000073(j) such that i>j}.

A113244 Prime differences of tetranacci numbers.

Original entry on oeis.org

2, 3, 7, 11, 13, 41, 79, 107, 179, 193, 293, 397, 769, 1489, 2099, 2843, 2857, 5507, 5521, 9181, 10463, 10663, 10667, 19079, 39619, 76423, 126743, 146539, 147283, 147311, 281081, 283949, 547229, 771073, 3919171, 3919543, 3919943, 7555879, 7555927, 10644589, 14564477

Offset: 1

Jonathan Vos Post, Oct 19 2005; corrected Oct 20 2005

A113188-A113194 deal with difference sets of Fibonacci numbers and Lucas numbers and primes in those difference sets. A113238-A113239 deal with the difference set of tribonacci numbers and primes in that difference set.

			a(1) = 2 because 4 - 2 = 2 where 4 and 2 are tetranacci numbers.
a(2) = 3 because 4 - 1 = 3 where 4 and 1 are tetranacci numbers.
a(3) = 7 because 8 - 1 = 7 where 8 and 1 are tetranacci numbers.
a(4) = 11 because 15 - 4 = 11 where 15 and 4 are tetranacci numbers.
a(5) = 13 because 15 - 2 = 13 where 15 and 2 are tetranacci numbers.

Cf. A000040, A000078, A113188-A113194, A113238, A113239, A113243.

isA113244 := proc(n)
    isprime(n) and isA113243(n) ;
end proc:
for n from 1 do
    p := ithprime(n) ;
    if isA113244(p) then
        printf("%d\n",p) ;
    end if;
end do: # R. J. Mathar, Oct 04 2014

{a(n)} = intersection of A000040 and A113243. {a(n)} = primes in the difference set of tetranacci sequence A000078, excluding prime tetranacci numbers A104535.

281081 inserted by R. J. Mathar, Oct 04 2014

A113189 Number of times that Fibonacci(n)-Fibonacci(i) is prime for i=0..n-3.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 0, 2, 1, 2, 3, 1, 1, 2, 0, 3, 3, 1, 0, 1, 1, 1, 0, 0, 2, 2, 0, 2, 5, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 2, 3, 1, 1, 2, 2, 0, 1, 1, 2, 2, 3, 7, 0, 0, 1, 0, 0, 1, 0, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 4, 0, 2, 6, 2, 2, 3, 0, 2, 1, 1, 0, 3, 0, 0, 6, 1, 0, 2, 5, 2, 0, 0, 1, 4, 2, 2

Offset: 3

T. D. Noe, Oct 17 2005

nonn

We exclude i=n-2 and i=n-1 because they yield Fibonacci(n-2) and Fibonacci(n-1), respectively. Sequence A113190 lists the n for which a(n)=0.

Cf. A113188 (primes that are the difference of two Fibonacci numbers).

Table[cnt=0; Do[If[PrimeQ[Fibonacci[n]-Fibonacci[i]], cnt++ ], {i, 0, n-3}]; cnt, {n, 3, 150}]

A113293 First differences of Lucas 3-step numbers.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 14, 18, 20, 28, 32, 36, 38, 50, 60, 64, 68, 70, 92, 110, 120, 124, 128, 130, 170, 202, 220, 230, 234, 238, 240, 312, 372, 404, 422, 432, 436, 440, 442, 574, 684, 744, 776, 794, 804, 808, 812, 814, 1056, 1258, 1368, 1428, 1460, 1478, 1488

Offset: 1

Jonathan Vos Post, Oct 23 2005

There are no primes in this sequence, except 2, as all values are odd, so all differences are even. Semiprimes include: a(3) = 4, a(4) = 6, a(6) = 10, a(7) = 14, a(13) = 38, a(26) = 202, a(35) = 422, a(44) = 794, a(54) = 1478, a(59) = 1942, a(66) = 2746, a(94) = 9326.

			a(0) = 0 because A001644(2)-A001644(0) = 3 - 3 = 0.
a(1) = 2 because A001644(2)-A001644(1) = 3 - 1 = 2.
a(2) = 4 because A001644(3)-A001644(2) = 7 - 3 = 4.
a(3) = 6 because A001644(3)-A001644(1) = 7 - 1 = 6.
a(75) = 5000 because A001644(14)-A001644(7) = 5071 - 71 = 5000.

Eric Weisstein's World of Mathematics, Lucas n-Step Number.
Noe, T. D. and Post, J. V., Primes in Fibonacci n-step and Lucas n-Step Sequences." J. Integer Seq. 8, Article 05.4.4, 2005.

Cf. A000040, A001358, A001644, A113188-A113194, A113238, A113239, A113244.

{a(n)} = { | A001644(i) - A001644(j) | such that i>=j}

A113190 Numbers n such that Fibonacci(n)-Fibonacci(i) is composite for all i=0..n-3.

Original entry on oeis.org

14, 22, 26, 30, 31, 34, 38, 40, 42, 44, 46, 54, 61, 62, 64, 65, 67, 78, 80, 82, 88, 92, 94, 95, 98, 102, 103, 109, 112, 113, 117, 119, 121, 122, 125, 126, 127, 134, 135, 138, 139, 142, 143, 152, 154, 155, 158, 166, 167, 170, 172, 174, 175, 176, 182, 188, 190, 193

Offset: 1

T. D. Noe, Oct 17 2005

nonn

These are the n such that A113189(n)=0.

Cf. A113188 (primes that are the difference of two Fibonacci numbers).

lst={}; Do[i=0; While[iHarvey P. Dale, Nov 05 2017 *)

A113294 First differences of Lucas 4-step numbers.

Original entry on oeis.org

1, 3, 4, 8, 11, 12, 19, 22, 23, 25, 36, 44, 47, 48, 73, 84, 92, 95, 96, 140, 165, 176, 184, 187, 188, 268, 316, 341, 352, 360, 363, 364, 517, 609, 657, 682, 693, 701, 704, 705, 998, 1174, 1266, 1314, 1339, 1350, 1358, 1361, 1362, 1923, 2264, 2440, 2532, 2580

Offset: 1

Jonathan Vos Post, Oct 23 2005

Lucas 4-step numbers are also known as "Tetranacci Lucas numbers" or "Tetranacci numbers with different initial conditions" in A073817. Primes in this sequence are A113295. In this sequence are: 13340261 = 11 * 19 * 29 * 31 * 71 is a product of 5 distinct 2-digit primes; 95550683 = 269 * 593 * 599 is a product of 3 distinct 3-digit primes.

			a(0) = 1 because A073817(0)-A001644(2) = 4 - 3 = 1.
a(1) = 3 because A073817(3)-A001644(0) = 7 - 4 = 3.
a(2) = 4 because A073817(3)-A001644(2) = 7 - 3 = 4.
a(3) = 8 because A073817(4)-A001644(3) = 15 - 7 = 8.
a(122) = 70000 because A073817(17)-A001644(3) = 70007 - 7 = 70000.

Eric Weisstein's World of Mathematics, Lucas n-Step Number.
Noe, T. D. and Post, J. V., Primes in Fibonacci n-step and Lucas n-Step Sequences." J. Integer Seq. 8, Article 05.4.4, 2005.

Cf. A000040, A073817, A113188-A113194, A113238, A113239, A113244, A113293, A113295.

{a(n)} = { | A073817(i) - A073817(j) | such that i>=j }

A113295 Prime differences of Lucas 4-step numbers.

Original entry on oeis.org

3, 11, 19, 23, 47, 73, 701, 1361, 4363, 9067, 9749, 17477, 18743, 18839, 36293, 70003, 116101, 134917, 366437, 465061, 498749, 501013, 1844033, 3590099, 13305307, 13341259, 13341619, 36229121, 49069367, 49570721, 95550661, 351427309

Offset: 1

Jonathan Vos Post, Oct 23 2005

These are primes from the sequence A113294, which is differences of Lucas 4-step numbers, also known as "Tetranacci Lucas numbers" or "Tetranacci numbers with different initial conditions" in A073817. Also in the difference set sequence are: 13340261 = 11 * 19 * 29 * 31 * 71 is a product of 5 distinct 2-digit primes; 95550683 = 269 * 593 * 599 is a product of 3 distinct 3-digit primes.

			a(1) = 3 because A073817(0)-A001644(1) = 4 - 1 = 3, a prime.
a(2) = 11 because A073817(4)-A001644(0) = 15 - 4 = 11, a prime.
a(3) = 19 because A073817(5)-A001644(3) = 26 - 7 = 19, a prime.
a(4) = 23 because A073817(5)-A001644(2) = 26 - 3 = 23, a prime.
a(16) = 70003 because A073817(17)-A001644(0) = 70007 - 4 = 70003, a prime.

Eric Weisstein's World of Mathematics, Lucas n-Step Number.
Noe, T. D. and Post, J. V., Primes in Fibonacci n-step and Lucas n-Step Sequences." J. Integer Seq. 8, Article 05.4.4, 2005.

Cf. A000040, A073817, A113188-A113194, A113238, A113239, A113244, A113293, A113294.

{a(n)} = Intersection of { | A073817(i) - A073817(j) | such that i>=j} and A000040. {a(n)} = Prime elements of { | A073817(i) - A073817(j) | such that i>=j}. {a(n)} = Prime elements of A113294.

cp's OEIS Frontend

A005479 Prime Lucas numbers (cf. A000032).

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A007298 Sums of consecutive Fibonacci numbers.

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A113239 Prime differences of tribonacci numbers.

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A113244 Prime differences of tetranacci numbers.

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A113189 Number of times that Fibonacci(n)-Fibonacci(i) is prime for i=0..n-3.

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A113293 First differences of Lucas 3-step numbers.

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A113190 Numbers n such that Fibonacci(n)-Fibonacci(i) is composite for all i=0..n-3.

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A113294 First differences of Lucas 4-step numbers.

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A113295 Prime differences of Lucas 4-step numbers.