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%I #8 Nov 28 2019 06:14:05
%S 1,2,4,8,9,10,25,71,103,115,157,231,329,1783,1835,4476,5128,12462,
%T 16274,25035,42174,72589,85968,147666,613726,1088825,1112415,3125316,
%U 3929736,5742036,7639447,25716100,32780150,48132247,76049401,100464259,108803364,186018939
%N The indexing sequence for successively better golden semiprimes.
%H Amiram Eldar, <a href="/A165569/b165569.txt">Table of n, a(n) for n = 1..48</a>
%F 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).
%t 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 *)
%o (MIT Scheme:)
%o (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))))))))
%o (define *phi* (/ (1+ (sqrt 5)) 2))
%Y The corresponding semiprimes are given by A165570(n) = A165571(n)*A165572(n).
%Y Cf. A108539.
%K nonn
%O 1,2
%A _Antti Karttunen_, Sep 22 2009
%E a(16)-a(38) from _Amiram Eldar_, Nov 28 2019
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