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A086386
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Numerators of the rational convergents to sqrt(3) if both numerators and denominators are primes.
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12
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OFFSET
| 1,1
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COMMENTS
| These numbers are rare.
General recurrence is a(n)=(a(1)-1)*a(n-1)-a(n-2), a(1)>=4, lim n->infinity a(n)= x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878, primes in it A121534. a(1)=5 gives A001834, primes in it A086386. a(1)=6 gives A030221, primes in it not in OEIS {29,139,3191,...}. a(1)=7 gives A002315, primes in it A088165. a(1)=8 gives A033890, primes in it not in OEIS (does there exist any ?). a(1)=9 gives A057080, primes in it not in OEIS {71,34649,16908641,...}. a(1)=10 gives A057081, primes in it not in OEIS {389806471,192097408520951,...}. [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), Sep 02 2008]
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PROG
| (PARI) cfracnum(m, f) = { cfr = vector(10000); x=f; for(n=0, m, i=floor(x); x=1/(x-i); cfr[n+1] = i; ); for(m1=0, m, r=cfr[m1+1]; forstep(n=m1, 1, -1, r = 1/r; r+=cfr[n]; ); numer=numerator(r); denom=denominator(r); if(isprime(numer) & isprime(denom), print1(numer", ")); ) }
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CROSSREFS
| Sequence in context: A149760 A149761 A149762 * A047155 A034548 A129166
Adjacent sequences: A086383 A086384 A086385 * A086387 A086388 A086389
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KEYWORD
| nonn,frac
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AUTHOR
| Cino Hilliard (hillcino368(AT)gmail.com), Sep 06 2003
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