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Search: id:A110881
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| A110881 |
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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 2 times the old bottom to get the new top. |
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1
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OFFSET
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1,2
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COMMENT
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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
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REFERENCES
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Prime Obsession, John Derbyshire, Joseph Henry Press, 2004, p. 16.
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FORMULA
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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)).
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EXAMPLE
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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.
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PROGRAM
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(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+.) }
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CROSSREFS
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Sequence in context: A098515 A059719 A066228 this_sequence A117443 A095858 A062792
Adjacent sequences: A110878 A110879 A110880 this_sequence A110882 A110883 A110884
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KEYWORD
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more,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Oct 02 2005, Jul 05 2007
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