A056816 -id:A056816 - cp's OEIS frontend

A231575 Indices of primes in A001590.

4, 5, 7, 9, 25, 29, 49, 79, 1613, 15205

Offset: 1

Robert Price, Nov 18 2013

a(11) > 2*10^5.

Cf. A001590, A231574, A000213, A056816, A157611.

v=[1,0,1]; for(n=4,1e4, if(ispseudoprime(t=v[1]+v[2]+v[3]), print1(n", ")); v=[v[2],v[3],t]) \\ Charles R Greathouse IV, Nov 18 2013

A001590(a(n)) = A231574(n). - Arthur O'Dwyer, 24 Jul 2024

Name clarified by Arthur O'Dwyer, Jul 24 2024

A231574 Primes in A001590.

Original entry on oeis.org

2, 3, 11, 37, 634061, 7256527, 1424681173049, 123937002926372177911

Offset: 1

Robert Price, Nov 18 2013

nonn

a(9) contains 427 digits and is too large to include here.

a(10) contains 4024 digits and is too large to include here.

Cf. A000213, A001590, A056816, A157611, A231575.

Name clarified by Arthur O'Dwyer, Jul 24 2024

A092836 Prime tribonacci numbers.

Original entry on oeis.org

2, 7, 13, 149, 19341322569415713958901, 15762679542071167858843489, 145082467753351661438130501937754420584096000083183992629

Offset: 1

Eric W. Weisstein, Mar 06 2004

nonn

The next term has 212 digits and is too large to fit. - Harvey P. Dale, Oct 08 2017

These primes are A000073(k) at k = 4, 6, 7, 11, 87, 98, 215, 802, 4202, ... The indices in A092835 are off by one. - R. J. Mathar, Dec 14 2011

Harry J. Smith, Table of n, a(n) for n = 1..8
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, Tribonacci Number

Cf. A000073, A092835, A056816.

Select[LinearRecurrence[{1,1,1},{0,0,1},1000],PrimeQ] (* Harvey P. Dale, Oct 08 2017 *)

t(m)= { if (m==0, return(a1)); if (m==1, return(a2)); if (m==2, return(a3)); r = a1 + a2 + a3; a1=a2; a2=a3; a3=r; return(r); } { a1=0; a2=1; a3=1; n=0; for (m=0, 1000, a=t(m); if (isprime(a), write("b092836.txt", n++, " ", a))); } \\ Harry J. Smith, Jun 21 2009

A157611 Numbers n such that the n-th tribonacci number as defined by A000213 is prime.

Original entry on oeis.org

3, 4, 6, 7, 10, 12, 13, 19, 23, 26, 32, 63, 95, 150, 324, 449, 1969, 2988, 6199, 8093, 23393, 25427, 34685, 35076, 44171, 45624, 83357, 116404

Offset: 1

Dmitry Kamenetsky, Mar 03 2009

nonn

The next term is greater than 10000.

a(29) > 2*10^5. - Robert Price, Nov 15 2013

Cf. A000213, A056816.

a(21)-a(28) from Robert Price, Nov 15 2013

A145027 a(n) = a(n-1) + a(n-2) + a(n-3) with a(1) = 2, a(2) = 3, a(3) = 4.

Original entry on oeis.org

2, 3, 4, 9, 16, 29, 54, 99, 182, 335, 616, 1133, 2084, 3833, 7050, 12967, 23850, 43867, 80684, 148401, 272952, 502037, 923390, 1698379, 3123806, 5745575, 10567760, 19437141, 35750476, 65755377, 120942994, 222448847, 409147218

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 30 2008

nonn

If the conjectured recurrence in A000382 is correct, then a(n) = A000382(n+2) - A000382(n+1), n>=4. - R. J. Mathar, Jan 30 2011

G. C. Greubel, Table of n, a(n) for n = 1..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000073, A000213, A001590, A003265, A056816, A081172, A001644.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1+x)*(2-x)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 22 2019

LinearRecurrence[{1,1,1},{2,3,4},33] (* Ray Chandler, Dec 08 2013 *)

my(x='x+O('x^30)); Vec(x*(1+x)*(2-x)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 22 2019

a=(x*(1+x)*(2-x)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Apr 22 2019

From R. J. Mathar, Jan 30 2011: (Start)
a(n) = -A000073(n-1) + A000073(n) + 2*A000073(n+1).
G.f. x*(1+x)*(2-x)/(1-x-x^2-x^3). (End)

A131354 Number of primes in the open interval between successive tribonacci numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 3, 5, 8, 12, 23, 38, 61, 109, 179, 312, 537, 920, 1598, 2779, 4835, 8461, 14784, 25984, 45696, 80505, 142165, 251300, 444930, 788828, 1400756, 2489594, 4430712, 7892037, 14073786, 25118167, 44869652, 80223172, 143535369, 257014148, 460524864, 825732764

Offset: 0

Jonathan Vos Post, Oct 21 2007

nonn

This is to tribonacci numbers A000073 as A052011 is to Fibonacci numbers and as A052012 is to Lucas numbers A000204. It is mere coincidence that all values until a(12) = 38 are themselves Fibonacci. The formula plus the known asymptotic prime distribution gives the asymptotic approximation of this sequence, which is the same even if we use one of the alternative definitions of tribonacci with different initial values.

			Between Trib(8)=24 and Trib(9)=44 we find the following primes: 29, 31, 37, 41, 43, hence a(8)=5.

Cf. A000073, A000720, A092836, A052011, A052012, A056816.

A131354 := proc(n)
    a := 0 ;
    for k from 1+A000073(n)  to A000073(n+1)-1 do
        if isprime(k) then
            a := a+1 ;
        end if;
    end do;
    a ;
end proc: # R. J. Mathar, Dec 14 2011

trib[n_] := SeriesCoefficient[x^2/(1 - x - x^2 - x^3), {x, 0, n}];
a[n_] := PrimePi[trib[n + 1] - 1] - PrimePi[trib[n]];
a /@ Range[0, 42] (* Jean-François Alcover, Apr 10 2020 *)

\\ here b(n) is A000073(n).
b(n)={polcoef(x^2/(1-x-x^2-x^3) + O(x*x^n), n)}
a(n)={primepi(b(n+1)-1) - primepi(b(n))} \\ Andrew Howroyd, Jan 02 2020

a(n) = A000720(A000073(n+1) - 1) - A000720(A000073(n)) for n >= 3. [formula edited Andrew Howroyd, Jan 02 2020]

Terms a(26) and beyond from Andrew Howroyd, Jan 02 2020

A156761 Tribonacci primes that are also primes when their digits are reversed.

Original entry on oeis.org

3, 5, 17, 31, 1201, 128199521

Offset: 1

Jonathan Vos Post, Feb 15 2009

This is to A115347 as Fibonacci numbers A000045 are to A000213 tribonacci numbers. Except for palidromatic tribonacci numbers {3, 5, ...} these are tribonacci emirps, A000213 INTERSECTION A006567.
a(7), if it exists, is greater than A000213(10000). - Dmitry Kamenetsky, Mar 03 2009
a(7), if it exists, is greater than A000213(116404), testing tribonacci primes using A157611. - Michael S. Branicky, May 27 2025

			a(5) = 1201 because 1201 is prime, and is the 13th tribonacci number A000213(13), and R(1201) = A004086(1201) = 1021 is also prime. a(6) = 128199521 = the 10th tribonacci prime A056816(10), and its digital reverse 125991821 is also prime.

Cf. A000040, A000045, A000213, A004086, A006567, A056816, A115347, A157611.

Select[LinearRecurrence[{1,1,1},{1,1,1},200],AllTrue[{#,IntegerReverse[ #]},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 31 2019 *)

{p such that p is in A000040 and p is in A000213 and R(p) = A004086(p) is in A000040} = {p such that p is in A056816 and R(p) = is in A000040}.

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A231575 Indices of primes in A001590.

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A231574 Primes in A001590.

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A092836 Prime tribonacci numbers.

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A157611 Numbers n such that the n-th tribonacci number as defined by A000213 is prime.

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A145027 a(n) = a(n-1) + a(n-2) + a(n-3) with a(1) = 2, a(2) = 3, a(3) = 4.

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A131354 Number of primes in the open interval between successive tribonacci numbers.

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A156761 Tribonacci primes that are also primes when their digits are reversed.

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