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A279369
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A mapping of rationals a/b (lowest form) to prime rationals p/q such that a/b = (p+1)/(q+1), where n (the sequence index) selects the rationals a/b from the triangle array A226314(n)/A054531(n) and a(n) selects the prime rationals p/q from the same array.
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0
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1, 12, 18, 58, 13, 74, 57, 19, 5, 72, 174, 178, 182, 429, 217, 138, 8, 225, 247, 272, 162, 825, 81, 83, 85, 849, 89, 999, 255, 1047, 23, 110, 484, 103, 1122, 288, 1383, 139, 114, 143, 1407, 32, 149, 1425, 1518, 408, 711, 176, 1677, 165, 727, 184, 1701, 188, 450, 906, 910, 914
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OFFSET
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1,2
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COMMENTS
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Rationals a/b (lowest form) can be mapped 1-to-1 to a positive integer n where a/b is the n-th term of the triangular array A226314(n)/A054531(n). Consider two function of x, f_1 = ax-1 and f_2 = bx-1. Then by Schinzel's Hypothesis H there are infinite values of x such that f_1 and f_2 are simultaneously prime allowing a/b to be expressed using two primes p and q as a/b=(p+1)/(q+1).
By choosing the least x for generating p=f_1 and q=f_2 (see A278635) it is possible to find a unique prime rational p/q that maps to rational a/b. If n is the sequence index that selects the rational a/b from the triangular array A226314(n)/A054531(n), then a(n) selects the prime rationals p/q from the same array.
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LINKS
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EXAMPLE
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a(7)=57 because A226314(7)/A054531(7)=1/4 and with least x=3 we have p=f_1=x-1=2 and q=f_2=4x-1=11. Therefore (p+1)/(q+1)=3/12=1/4. Also A226314(57)/A054531(57)=p/q=2/11.
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MATHEMATICA
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func[{i_, j_}] := {j(j-1)/2+i->{j+(i-j)/GCD[i, j], j/GCD[i, j]}}; rfunc[{i_, j_}] := {{j+(i-j)/GCD[i, j], j/GCD[i, j]}->j(j-1)/2+i}; getx[{a_, b_}] := Module[{f1, f2, x}, If[a==b, {1, 1}, (f1=a*x-1; f2=b*x-1; x=1; While[(!PrimeQ[f1]||!PrimeQ[f2])&&x<10^5, x++]; If[x==10^5, Abort[], {f1, f2}])]]; assoc=Association@Flatten[Table[func[{a, b}], {b, 1000}, {a, b}], 1]; rassoc=Association@Flatten[Table[rfunc[{a, b}], {b, 1000}, {a, b}], 1]; Table[rassoc[getx[assoc[n]]], {n, 1, 100}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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