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A146348 Primes p such that continued fraction of (1+sqrt(p))/2 has period 3. 37
17, 37, 61, 101, 197, 257, 317, 401, 461, 557, 577, 677, 773, 1129, 1297, 1429, 1601, 1877, 1901, 2917, 3137, 4357, 4597, 5417, 5477, 6053, 7057, 8101, 8761, 8837, 10733, 11621, 12101, 13457, 13877, 14401, 15277, 15377, 15877, 16333, 16901, 17737, 17957, 18329, 21317, 22501, 23593, 24337, 25601, 28901, 30137, 30977, 32401, 33857, 41453, 41617, 42437, 44101 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes in A146328. Finite A050952 is subset of this sequence.

From Michel Lagneau, Sep 03 2014: (Start)

The primes of the form p = n^2+1 for n>2 are in the sequence, and the continued fraction of (1+sqrt(p))/2 is [n/2; 1, 1, n-1, 1, 1, n-1, 1, 1,...] with the period (1, 1, n-1).

We observe that the other primes {61, 317, 461, 557, 773, 1129, 1429,…} are prime divisors of composites numbers of the form k^2+1 where k = 11, 114, 48, 118, 317, 168, 620,… .

(End)

Another possibly infinite subset of the sequence is primes of the form 100*k^2-44*k+5, where the continued fraction is [5*k-1; 2, 2, 10*k-3, ...] with period [2, 2, 10*k-3].  This includes {61, 317, 773, 1429, 4597, 6053, ...}. - Robert Israel, Sep 03 2014

LINKS

Zak Seidov, Table of n, a(n) for n = 1..200

MAPLE

A146326 := proc(n) if not issqr(n) then numtheory[cfrac]( (1+sqrt(n))/2, 'periodic', 'quotients') ; nops(%[2]) ; else 0 ; fi; end: isA146348 := proc(n) RETURN(isprime(n) and A146326(n) = 3) ; end: for n from 2 to 4000 do if isA146348(n) then printf("%d, \n", n) ; fi; od: # R. J. Mathar, Sep 06 2009

MATHEMATICA

okQ[n_] := Length[ContinuedFraction[(1 + Sqrt[n])/2][[2]]] == 3; Select[Prime[Range[100]], okQ]

CROSSREFS

Cf. A000290, A050952, A078370, A146326-A146345, A146348-A146360, A146363.

Sequence in context: A146328 A161549 A269788 * A050952 A256517 A323603

Adjacent sequences:  A146345 A146346 A146347 * A146349 A146350 A146351

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 30 2008

EXTENSIONS

1019 removed; more terms added by R. J. Mathar, Sep 06 2009

More terms from Zak Seidov, Mar 09 2011

STATUS

approved

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Last modified July 18 07:10 EDT 2019. Contains 325134 sequences. (Running on oeis4.)