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 1-4 of 4 results.

A286682 a(n) = A059784(n+1) - A059784(n)^2.

Original entry on oeis.org

1, 4, 12, 4, 22, 12, 114, 4, 138, 142, 2956, 6388, 5248, 17532, 96930, 83782, 1464, 897448, 300832, 26908
Offset: 1

Views

Author

Jeppe Stig Nielsen, May 12 2017

Keywords

Comments

This sequence relates to A059784 just like A108739 relates to the Mills primes A051254.
That this leads to a Mills-like real constant C such that floor(C^2^n) is a prime number for any natural number n, requires a proof of Legendre's conjecture that there is always a prime between consecutive perfect squares.
a(18) and a(19) generate 96042- and 192083-decimal digit probable primes. - Serge Batalov, May 27 2024
a(20) generates a 384166-decimal digit probable prime. - Serge Batalov, May 27 2024

Examples

			A059784(8) by construction can be written ((((((2^2 + 1)^2 + 4)^2 + 12)^2 + 4)^2 + 22)^2 + 12)^2 + 114. Taking out the addends gives 1, 4, 12, 4, 22, 12, 114 which lists the first seven terms of this sequence.

Crossrefs

Cf. A059784, A108739, A051254.

Programs

  • Mathematica
    Map[#2 - #1^2 & @@ # &, Partition[NestList[NextPrime[#^2] &, 2, 12], 2, 1]] (* Michael De Vlieger, May 12 2017 *)
  • PARI
    p=2;while(1,a=nextprime(p^2);print1(a-p^2,", ");p=a)

Extensions

a(14)-a(17) from Serge Batalov, May 26 2024
a(18)-a(20) from Serge Batalov, May 27 2024

A059785 a(n+1) = prevprime(a(n)^2). Largest prime prior to the square of previous prime. Initial value = 2.

Original entry on oeis.org

2, 3, 7, 47, 2207, 4870843, 23725111530599, 562880917139361624513298747, 316834926879648887020732217199607668221645859671769857
Offset: 1

Views

Author

Labos Elemer, Feb 22 2001

Keywords

Comments

The next term is too large to show here - see the b-file.

Links

Crossrefs

Cf. A006992, A055496, A005384, A005385, A059784.

Programs

  • Mathematica
    NestList[NextPrime[#^2,-1]&,2,10] (* Harvey P. Dale, Jan 16 2016 *)

Extensions

Offset and some values corrected by Franklin T. Adams-Watters, Jul 30 2009

A112597 Decimal expansion of x, where x is the smallest number for which floor(x^(2^y)) is prime for every y > 0 (assuming the truth of Legendre's conjecture).

Original entry on oeis.org

1, 5, 2, 4, 6, 9, 9, 9, 6, 0, 5, 3, 8, 0, 9, 4, 3, 5, 9, 9, 2, 3, 3, 6, 3, 5, 7, 5, 6, 8, 8, 4, 2, 1, 1, 6, 2, 2, 2, 0, 2, 2, 3, 6, 2, 3, 1, 9, 9, 7, 7, 1, 2, 1, 9, 8, 4, 5, 7, 2, 2, 2, 6, 5, 6, 8, 4, 1, 6, 3, 0, 4, 8, 2, 0, 2, 3, 5, 7, 2, 3, 2, 3, 7, 0, 4, 8, 2, 3, 8, 3, 0, 4, 8, 6, 0, 9, 5, 7, 9, 5, 9, 9, 7, 6
Offset: 1

Views

Author

Martin Raab, Dec 21 2005

Keywords

Examples

			1.524699960538...
Repeated squaring gives the primes 2, 5, 29, 853, 727613, 529420677791, 280286254072681840639693, ... (A059784).

Crossrefs

Cf. A059784, A051021, decimal expansion of Mills' constant where floor(x^(3^y)) is prime for every y > 0.

Extensions

Name clarified by Thomas Scheuerle, Mar 28 2025

A382261 a(n) = floor(x^(phi^n)), where phi = (1+sqrt(5))/2 and x is the constant A382260.

Original entry on oeis.org

2, 3, 7, 23, 163, 3803, 620549, 2359981439, 1464484123012601, 3456155348019933976288373, 5061484633840283809323162088349619180781, 17493277186167814180104995425523045477935447066389138909089293633
Offset: 1

Views

Author

Thomas Scheuerle, Mar 19 2025

Keywords

Comments

Conjecture: All terms are prime numbers. For details see A382260.

Crossrefs

Cf. A001622, A059784, A051254, A243358, A382260. A090253.
Cf. A090253 ( similar growth ).

Formula

nextprime(a(n-2)*a(n-1)) <= a(n) < nextprime((a(n-2)+1)*a(n-1)).
Showing 1-4 of 4 results.

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