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A165569
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The indexing sequence for successively better golden semiprimes.
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4
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1, 2, 4, 8, 9, 10, 25, 71, 103, 115, 157, 231, 329, 1783, 1835, 4476, 5128, 12462, 16274, 25035, 42174, 72589, 85968, 147666, 613726, 1088825, 1112415, 3125316, 3929736, 5742036, 7639447, 25716100, 32780150, 48132247, 76049401, 100464259, 108803364, 186018939
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OFFSET
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1,2
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LINKS
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FORMULA
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a(1)=1, and for n>1, a(n) = first such i>a(n-1) that abs(phi - A108539(i)/A000040(i)) < abs(phi - A108539(a(n-1))/A000040(a(n-1))), where phi = (1+sqrt(5))/2 (Golden ratio).
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MATHEMATICA
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f[p_] := Module[{x = GoldenRatio * p, p1, p2}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; If[p2 - x > x - p1, p1, p2]]; seq={}; k=0; dm = 1; p1 = 1; Do[p1 = NextPrime[p1]; k++; p2 = f[p1]; d = Abs[p2/p1 - GoldenRatio]; If[d < dm, dm = d; AppendTo[seq, k]], {10^4}]; seq (* Amiram Eldar, Nov 28 2019 *)
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PROG
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(MIT Scheme:)
(define (A165569 n) (if (= 1 n) 1 (let* ((i (A165569 (-1+ n))) (champion (abs (- *phi* (/ (A108539 i) (A000040 i)))))) (let loop ((i (1+ i))) (cond ((< (abs (- *phi* (/ (A108539 i) (A000040 i)))) champion) i) (else (loop (1+ i))))))))
(define *phi* (/ (1+ (sqrt 5)) 2))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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