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.

A099000 Indices k such that the k-th prime is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 6, 24, 51, 251, 3121, 42613, 23023556, 143130479, 2602986161967491
Offset: 1

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Author

Rick L. Shepherd, Nov 06 2004

Keywords

Comments

From Hugo Pfoertner, Jan 06 2020: (Start)
The computation of the next two terms, corresponding to the primes F(131) = A005478(13) = 1066340417491710595814572169, and F(137) = A005478(14) = 19134702400093278081449423917, should already be within reach with current (2020) technology, e.g. with Kim Walisch's "primecount" program, which allows massive parallelization. An exact determination of the following term a(15), which corresponds to F(359), is beyond any imaginable technical possibility.
Estimates for a(13)-a(15), found by using the PARI program from A121046 in a bisection loop, with an accuracy that corresponds to the shown number of digits, are as follows:
a(13) = primepi(F(131)) ~= 1.741898800848...*10^25,
a(14) = primepi(F(137)) ~= 2.9848914766265...*10^26,
a(15) = primepi(F(359)) ~= 2.78114064956041656819790214151422895...*10^72.
(End)

Crossrefs

Cf. A001605 (n-th Fibonacci number is prime), A005478 (Prime Fibonacci numbers).
Cf. A121046.

Programs

  • Mathematica
    PrimePi[Select[Fibonacci[Range[80]], PrimeQ]]
  • PARI
    print1("1, 2");forprime(p=5,47,if(isprime(fibonacci(p)),print1(", "primepi(fibonacci(p))))) \\ Charles R Greathouse IV, Aug 21 2011

Formula

a(n) = A000720(A005478(n)). - M. F. Hasler, Aug 21 2011

Extensions

a(11) from Ryan Propper, Oct 16 2005
a(12) from Charles R Greathouse IV, Aug 21 2011

A095124 a(n) = prime(2^(2^n)).

Original entry on oeis.org

3, 7, 53, 1619, 821641, 104484802057, 870566678511500413493
Offset: 0

Views

Author

Cino Hilliard, Jun 19 2004

Keywords

Comments

An estimate using the program in A121046 for a(7) is 3.13863199556707166...*10^40. - Hugo Pfoertner, Mar 26 2024

Crossrefs

Programs

  • Mathematica
    Table[Prime[2^(2^n)],{n,0,5}] (* Harvey P. Dale, Oct 13 2013 *)
  • PARI
    f(n) = for(x=0,n,print1(prime(2^2^x)","))

Formula

a(n) = A000040(A001146(n)). - Michel Marcus, Mar 25 2024
a(n) = A033844(2^n). - Rainer Rosenthal, Mar 25 2024

Extensions

a(6) from Rainer Rosenthal, Mar 25 2024

A123233 Difference between the (10^n)-th prime and the Riemann-Gram approximation of the (10^n)-th prime.

Original entry on oeis.org

1, 0, 5, -4, -39, -24, 1823, -6566, -1844, -34087, 84846, -449836, -1117632, -3465179, -1766196, -11290074, 105510354, -208774399, 704933861
Offset: 0

Views

Author

Cino Hilliard, Oct 06 2006

Keywords

Comments

The algorithm in the PARI script below produces the 10^n-th prime accurate to first n/2 places. Conjecture: The sign of the terms in this sequence changes infinitely often. Based on the small sample presented here, it appears the negative terms occur much more often.

Examples

			a(1) = prime(10) - primeGR(10) = 29 - 29 = 0.
		

Crossrefs

Programs

  • PARI
    primeGR(n) =
    \\ A good approximation for the n-th prime number using
    \\ the Gram-Riemann approximation of Pi(x)
    { local(x,px,r1,r2,r,p10,b,e); b=10; p10=log(n)/log(10); if(Rg(b^p10*log(b^(p10+1)))< b^p10,m=p10+1,m=p10); r1 = 0; r2 = 7.18281828; for(x=1,400, r=(r1+r2)/2; px = Rg(b^p10*log(b^(m+r))); if(px <= b^p10,r1=r,r2=r); r=(r1+r2)/2; ); floor(b^p10*log(b^(m+r))+.5); }
    Rg(x) =
    \\ Gram's Riemann's Approx of Pi(x)
    { local(n=1,L,s=1,r); L=r=log(x); while(s<10^40*r, s=s+r/zeta(n+1)/n; n=n+1; r=r*L/n); (s) }

Formula

prime(10^x)-primeRG(10^x), where prime(n) is the n-th prime and primeRG(n)is an approximation of the n-th prime number based on an exponential bisection routine that uses the Riemann-Gram approximation of Pi(x). The flow of the routine is evident in the PARI script below.
a(n) = A006988(n) - A121046(n) for n >= 1. - Amiram Eldar, Jul 04 2024

Extensions

a(17)-a(18) from Amiram Eldar, Jul 04 2024
Showing 1-3 of 3 results.