A110881 Starting a priori with the fraction 1/1, list n when the numerator and denominator are both prime for fractions built according to the rule: Add old top and old bottom to get the new bottom, add old top and twice the old bottom to get the new top.

1, 2, 4, 28, 58

Offset

1,2

Comments

k is the multiple 4 in the PARI code. The sequence of fractions found with the property that both numerator and denominator are prime is as follows.

n, num/denom

1, 3/2

2, 7/5

4, 41/29

28, 63018038201/44560482149

58, 19175002942688032928599/13558774610046711780701

References

Prime Obsession, John Derbyshire, Joseph Henry Press, 2004, p. 16.

Links

Table of n, a(n) for n=1..5.

Formula

Given a(0)=1,b(0)=1 then for i=1,2,.. a(i)/b(i) = (a(i-1)+2*b(i-1))/(a(i-1)+b(i-1)).

Example

For k=2, n = 1, we start with fraction 1/1. The new bottom is old top + old bottom = 1+1 = 2. Then we add 1 (old top) + 2*1 (old bottom) to get 3, the new top. So our first fraction is 3/2. Since these are both prime, n=1 is the first entry. Continuing with this fraction 3/2, we repeat the rule. The new bottom is 3+2 = 5. Then we add 3 + 2*2 to get 7, the new top. So our second fraction is 7/5. Since both numerator and denominator are prime, n=2 is the second entry.

Prog

(PARI) primenumdenom(n, k) = { local(a, b, x, tmp, v); a=1; b=1; for(x=1, n, tmp=b; b=a+b; a=k*tmp+a; if(tmp=1, v=a, v=b); if(ispseudoprime(a)&ispseudoprime(b), print1(x", "); ) ); print(); print(a/b+.) }

Crossrefs

Sequence in context: A323447 A227303 A066228 * A156449 A259375 A192374

Adjacent sequences: A110878 A110879 A110880 * A110882 A110883 A110884

Keyword

more,nonn

Author

Cino Hilliard, Oct 02 2005, Jul 05 2007

Status

approved