cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A201001 Primes in the Lucas U(3,-1) sequence.

Original entry on oeis.org

3, 109, 141481, 183642229, 12435170468929559540592573684601
Offset: 1

Views

Author

R. J. Mathar, Jan 08 2013

Keywords

Comments

The Lucas U(3,-1) sequence is A006190. This sequence here contains U(3,-1,n) at n = 2, 5, 11, 17, 61, 103, 167,..., listed in A209493.

Links

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

Views

Author

Vaclav Kotesovec, Sep 08 2013

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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[[#]]]&]

A305534 Index of the smallest prime in the n-Fibonacci sequence, or the Lucas U(n,-1) sequence.

Original entry on oeis.org

3, 2, 2, 3, 2, 3, 2, 5, 29, 3, 2, 5, 2, 3, 23, 3, 2, 7, 2, 3, 29, 19, 2, 3, 83, 3, 53, 19, 2, 5, 2, 5, 5, 5479, 71, 3, 2, 17, 11, 3, 2, 37, 2, 31, 5, 11, 2, 5
Offset: 1

Views

Author

Eric Chen, Jun 04 2018

Keywords

Comments

Smallest k such that the k-th Fibonacci polynomial evaluated at x=n is prime. (The first few Fibonacci polynomials are 1, x, x^2 + 1, x^3 + 2*x, x^4 + 3*x^2 + 1, x^5 + 4*x^3 + 3*x, ...)
All terms are primes, since if a divides b, then the a-th term of the n-Fibonacci sequence also divides the b-th term of the n-Fibonacci sequence.
Corresponding primes are 2, 2, 3, 17, 5, 37, 7, 4289, 726120289954448054047428229, 101, 11, 21169, 13, 197, 82088569942721142820383601, 257, 17, 34539049, 19, 401, ...
a(n) = 2 if and only if n is prime.
a(n) = 3 if and only if n^2 + 1 is prime (A005574), except n=2 (since 2 is the only prime p such that p^2 + 1 is also prime).
a(34) > 1024, does a(n) exist for all n >= 1? (However, 17 is the only prime in the first 1024 terms of the 4-Fibonacci sequence, and it seems that 17 is the only prime in the 4-Fibonacci sequence.)
a(35)..a(48) = 71, 3, 2, 17, 11, 3, 2, 37, 2, 31, 5, 11, 2, 5, a(50)..a(54) = 11, 11, 23, 2, 3, a(56) = 3, a(58)..a(75) = 5, 2, 47, 2, 5, 311, 13, 233, 3, 2, 5, 11, 5, 2, 7, 2, 3, 5. Unknown terms a(34), a(49), a(55), a(57), exceed 1024, if they exist.
a(49) > 20000, if it exists. - Giovanni Resta, Jun 06 2018

Crossrefs

Cf. A001605, A096650, A209493, which are the indices of the primes in the n-Fibonacci sequence for n = 1, 2, 3.
Cf. A005478, A086383, A201001, which are the primes in the n-Fibonacci sequence for n = 1, 2, 3.
Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A041041, A049666, A041061 (the n-Fibonacci sequence for n = 1 to 12).
Cf. A302990 (for n-step Fibonacci sequence instead of n-Fibonacci sequence).

Programs

  • PARI
    b(n,k)=([n,1;1,0]^k)[1,2]
a(n)=for(k=1,2^12,if(ispseudoprime(b(n,k)),return(k)))

Extensions

a(34)-a(48) from Giovanni Resta, Jun 06 2018
Showing 1-3 of 3 results.

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