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.

A247485 Integer part of 2*sqrt(prime(n)) + 1.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 21, 21, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 31, 31, 32, 32, 33, 33, 33, 33, 34, 34, 34, 35, 36, 36, 36, 36, 37
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 20 2014

Keywords

Comments

A117767(n) = 2*floor(sqrt(prime(n))) + 1 <= a(n);
a(A247514(n)) = A117767(A247514(n)); a(A247515(n)) > A117767(A247515(n)).
Andrica's conjecture: a(n) <= A001223(n).

Links

Crossrefs

Cf. A117767, A001223, A000040, A000006, A247514, A247515.

Programs

  • Haskell
    a247485 = (+ 1) . floor . (* 2) . sqrt . fromIntegral . a000040
  • Mathematica
    Floor[2Sqrt[Prime[Range[70]]]+1] (* Harvey P. Dale, Sep 04 2020 *)
  • PARI
    a(n) = 1+sqrtint(4*prime(n)); \\ Michel Marcus, Aug 26 2021

A096494 Largest value in the periodic part of the continued fraction of sqrt(prime(n)).

Original entry on oeis.org

2, 2, 4, 4, 6, 6, 8, 8, 8, 10, 10, 12, 12, 12, 12, 14, 14, 14, 16, 16, 16, 16, 18, 18, 18, 20, 20, 20, 20, 20, 22, 22, 22, 22, 24, 24, 24, 24, 24, 26, 26, 26, 26, 26, 28, 28, 28, 28, 30, 30, 30, 30, 30, 30, 32, 32, 32, 32, 32, 32, 32, 34, 34, 34, 34, 34, 36, 36, 36, 36, 36, 36
Offset: 1

Views

Author

Labos Elemer, Jun 29 2004

Keywords

Examples

			n=31: prime(31) = 127, and the periodic part is {3,1,2,2,7,11,7,2,2,1,3,22}, so a(31)=22.

Links

Crossrefs

Cf. A000006, A003285, A005980, A054269.
Cf. A096491, A096492, A096493, A096495, A096496.
Cf. A117767.

Programs

  • Haskell
    a096494 = (* 2) . a000006  -- Reinhard Zumkeller, Sep 20 2014
  • Maple
    A096491 := proc(n)
if issqr(n) then
sqrt(n) ;
else
numtheory[cfrac](sqrt(n),'periodic','quotients') ;
%[2] ;
max(op(%)) ;
end if;
end proc:
A096494 := proc(n)
option remember ;
A096491(ithprime(n)) ;
end proc: # R. J. Mathar, Mar 18 2010
  • Mathematica
    {te=Table[0, {m}], u=1}; Do[s=Max[Last[ContinuedFraction[Prime[n]^(1/2)]]]; te[[u]]=s;u=u+1, {n, 1, m}];te
a[n_]:=IntegerPart[Sqrt[Prime[n]]] 2 IntegerPart[Sqrt[#]]&/@Prime[Range[90]] (* Vincenzo Librandi, Aug 09 2015 *)

Formula

It seems that lim_{n->infinity} a(n)/n = 0. - Benoit Cloitre, Apr 19 2003
a(n) = 2*A000006(n). - Benoit Cloitre, Apr 19 2003

A247514 Numbers k such that 2*floor(sqrt(prime(k))) = floor(2*sqrt(prime(k))).

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 16, 19, 20, 23, 24, 26, 27, 28, 29, 31, 32, 35, 36, 40, 41, 42, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 62, 67, 68, 73, 74, 75, 79, 80, 81, 86, 87, 88, 89, 93, 94, 95, 96, 100, 101, 106, 107, 108, 109, 115, 116, 117, 118, 123, 124
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 20 2014

Keywords

Comments

A117767(a(n)) = A247485(a(n)); complement of A247515.

Links

Crossrefs

Cf. A117767, A247485, A000196, A000040, A000006, A247515.

Programs

  • Haskell
    a247514 n = a247514_list !! (n-1)
a247514_list = filter (\x -> a117767 x == a247485 x) [1..]
  • Mathematica
    A247514Q[k_]:=With[{r=Sqrt[Prime[k]]},2Floor[r]==Floor[2r]];
Select[Range[200],A247514Q] (* Paolo Xausa, Oct 23 2023 *)
  • PARI
    isok(k) = my(p=prime(k)); 2*sqrtint(p) == sqrtint(4*p); \\ Michel Marcus, Apr 29 2023

A247515 Numbers k such that 2*floor(sqrt(prime(k))) < floor(2*sqrt(prime(k))).

Original entry on oeis.org

2, 4, 6, 9, 11, 14, 15, 17, 18, 21, 22, 25, 30, 33, 34, 37, 38, 39, 43, 44, 47, 48, 53, 54, 59, 60, 61, 63, 64, 65, 66, 69, 70, 71, 72, 76, 77, 78, 82, 83, 84, 85, 90, 91, 92, 97, 98, 99, 102, 103, 104, 105, 110, 111, 112, 113, 114, 119, 120, 121, 122, 127
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 20 2014

Keywords

Comments

A117767(a(n)) < A247485(a(n)); complement of A247514.

Links

Crossrefs

Cf. A117767, A247485, A000196, A000040, A000006, A247514.

Programs

  • Haskell
    a247515 n = a247515_list !! (n-1)
a247515_list = filter (\x -> a117767 x < a247485 x) [1..]
  • Mathematica
    A247515Q[k_]:=With[{r=Sqrt[Prime[k]]},2Floor[r]A247515Q] (* Paolo Xausa, Oct 23 2023 *)
Showing 1-4 of 4 results.

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