A096493 -id:A096493 - cp's OEIS frontend

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

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

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)=22.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

a096494 = (* 2) . a000006  -- Reinhard Zumkeller, Sep 20 2014

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

{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 *)

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

A096492 Number of distinct terms in continued fraction period of square root of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 4, 2, 3, 4, 3, 2, 1, 1, 2, 3, 3, 2, 4, 2, 3, 3, 2, 1, 1, 2, 2, 2, 2, 2, 4, 3, 3, 5, 3, 2, 1, 1, 2, 4, 3, 4, 2, 2, 3, 2, 4, 3, 5, 3, 2, 1, 1, 2, 5, 2, 4, 3, 4, 2, 3, 2, 2, 5, 4, 3, 3, 2, 1, 1, 2, 2, 3, 4, 2, 3, 3, 2, 3, 4, 4, 6, 3, 3, 3, 3, 2, 1, 1, 2, 5, 2, 2

Offset: 1

Labos Elemer, Jun 29 2004

nonn

Essentially the same as A028832. - Amiram Eldar, Nov 10 2021

			n=127: the period={3,1,2,2,7,11,7,2,2,1,3,22},distinct-terms={1,2,3,7,11,22}, so a[127]=6;

Cf. A003285, A013646, A028832, A096491, A096493.

{tc=Table[0, {m}], u=1}; Do[s=Length[Union[Last[ContinuedFraction[n^(1/2)]]]]; tc[[u]]=s;u=u+1, {n, 1, m}], tc

a(n) = 1 if n is a square and a(n) = A028832(n) otherwise. - Amiram Eldar, Nov 10 2021

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 *)

cp's OEIS Frontend

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

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A096492 Number of distinct terms in continued fraction period of square root of n.

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

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