A096494 -id:A096494 - cp's OEIS frontend

A117767 a(n) is the difference between the smallest square greater than prime(n) and the largest square less than prime(n), where prime(n) = A000040(n) is the n-th prime number.

3, 3, 5, 5, 7, 7, 9, 9, 9, 11, 11, 13, 13, 13, 13, 15, 15, 15, 17, 17, 17, 17, 19, 19, 19, 21, 21, 21, 21, 21, 23, 23, 23, 23, 25, 25, 25, 25, 25, 27, 27, 27, 27, 27, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 33, 33, 33, 33, 33, 33, 33, 35, 35, 35, 35, 35, 37, 37, 37, 37, 37, 37

Offset: 1

Odimar Fabeny, Apr 15 2006

From Reinhard Zumkeller, Sep 20 2014: (Start)
a(n) <= floor(2*sqrt(prime(n))) + 1 = A247485(n).
a(A247514(n)) = A247485(A247514(n)).
a(A247515(n)) < A247485(A247515(n)). (End)

			The 7th prime number is 17, which is between the consecutive squares 16 and 25, so a(7) = 25 - 16 = 9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Legendre's Conjecture.

Cf. A000040, A000006.

Cf. A096494, A247485, A247514, A247515.

a117767 = (+ 1) . (* 2) . a000006  -- Reinhard Zumkeller, Sep 20 2014

Mathematica
```
a[n_]:=2Floor[Sqrt[Prime[n]]]+1
```

{ forprime(p=2,200, f = floor(sqrt(p)) ; print1(2*f+1,",") ; ) ; } \\ R. J. Mathar, Apr 21 2006

a(n) = 2*A000006(n) + 1.

a(n) = 2*floor(sqrt(prime(n))) + 1. - R. J. Mathar, Apr 21 2006

More terms from R. J. Mathar, Apr 21 2006

Edited by Dean Hickerson, Jun 03 2006

A096495 Number of distinct terms in the periodic part of the continued fraction for sqrt(prime(n)).

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 1, 4, 3, 3, 4, 1, 2, 4, 3, 3, 4, 5, 5, 4, 3, 3, 2, 3, 3, 1, 5, 4, 6, 3, 6, 4, 3, 6, 5, 7, 5, 6, 3, 3, 6, 6, 6, 5, 1, 7, 8, 3, 2, 3, 3, 6, 5, 5, 1, 4, 2, 7, 7, 5, 6, 3, 6, 6, 6, 5, 8, 6, 5, 4, 4, 3, 7, 3, 9, 4, 3, 7, 1, 6, 6, 8, 7, 6, 3, 2, 5, 7, 5, 9, 4, 6, 9, 8, 4, 4, 6, 6, 8, 9, 8, 2, 4, 6, 10

Offset: 1

Labos Elemer, Jun 29 2004

nonn

			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) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003285, A028832, A054269, A005980, A096491, A096492, A096493, A096494, A096496.

{te=Table[0, {m}], u=1}; Do[s=Length[Union[Last[ContinuedFraction[Prime[n]^(1/2)]]]]; te[[u]]=s;u=u+1, {n, 1, m}];te
Table[Length[Union[ContinuedFraction[Sqrt[Prime[n]]][[2]]]],{n,110}] (* Harvey P. Dale, Jun 22 2017 *)

a(n) = A028832(A000040(n)). - Amiram Eldar, Nov 10 2021

A096496 Number of distinct primes in the periodic part of the continued fraction for sqrt(prime(n)).

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 2, 0, 1, 2, 1, 1, 2, 2, 3, 2, 1, 1, 0, 2, 1, 0, 1, 1, 2, 1, 4, 2, 1, 4, 2, 4, 3, 4, 1, 0, 4, 1, 3, 2, 0, 3, 4, 1, 0, 1, 1, 2, 2, 2, 0, 0, 1, 1, 3, 1, 1, 0, 4, 3, 3, 1, 5, 3, 2, 2, 2, 1, 3, 2, 4, 2, 1, 2, 0, 3, 4, 5, 5, 3, 1, 0, 3, 4, 1, 4, 1, 3, 3, 2, 1, 1, 2, 2, 2, 4, 4, 0, 2, 3, 4

Offset: 1

Labos Elemer, Jun 29 2004

nonn

			n=31: prime(31) = 127, and the periodic part of the continued fraction of sqrt(127) is {3,1,2,2,7,11,7,2,2,1,3,22}, so a(31) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003285, A054269, A005980, A096491, A096492, A096493, A096494, A096495.

{te=Table[0, {m}], u=1}; Do[s=Count[PrimeQ[Union[Last[ContinuedFraction[f[n]^(1/2)]]]], True]; te[[u]]=s;u=u+1, {n, 1, m}];te
Count[Union[ContinuedFraction[Sqrt[#]][[2]]],?PrimeQ]&/@Prime[ Range[ 110]] (* _Harvey P. Dale, Apr 27 2016 *)

A308720 The maximum value in the continued fraction of sqrt(n), or 0 if there is no fractional part.

Original entry on oeis.org

0, 0, 2, 2, 0, 4, 4, 4, 4, 0, 6, 6, 6, 6, 6, 6, 0, 8, 8, 8, 8, 8, 8, 8, 8, 0, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 0, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 0, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 0, 16, 16, 16, 16, 16, 16, 16

Offset: 0

Karl Fischer, Jun 19 2019

The continued fraction expansion of sqrt(n) is periodic, and the maximal element is the last element in the period, 2*floor(sqrt(n)).

Oskar Perron, Die Lehre von den Kettenbrüchen, B. G. Teubner (1913), section 24, p. 87.

Cf. A000196, A003285, A096494.

{0} ~Join~ Table[2 Mod[Floor@ Sqrt@ n, Ceiling@ Sqrt@ n], {n, 100}] (* Giovanni Resta, Jun 29 2019 *)

a(k^2) = 0.
a(m) = 2 * floor(sqrt(m)) for nonsquare m.
a(n) = 2 * A320471(n) for n > 0.

More terms from Giovanni Resta, Jun 29 2019

cp's OEIS Frontend

A117767 a(n) is the difference between the smallest square greater than prime(n) and the largest square less than prime(n), where prime(n) = A000040(n) is the n-th prime number.

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A096495 Number of distinct terms in the periodic part of the continued fraction for sqrt(prime(n)).

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Mathematica

Formula

A096496 Number of distinct primes in the periodic part of the continued fraction for sqrt(prime(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A308720 The maximum value in the continued fraction of sqrt(n), or 0 if there is no fractional part.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions