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%I #17 Nov 28 2019 06:14:13
%S 6,15,77,589,851,1363,15229,201563,512893,644251,1366553,3416003,
%T 7881197,377331139,400711231,2963563859,4035221017,28862500577,
%U 52027213697,133793658289,418298061641,1363588753103,1970239102459,6355462656397,136388198153719,465337655023099
%N Successively better golden semiprimes.
%C This is lexicographically earliest sequence of such semiprimes p*q, starting from 6=2*3, that for each successive term p*q, q/p is a better approximant of Golden ratio (1+sqrt(5))/2 than the previous term. See A165569 for the exact procedure.
%C Can it be proved that this a subset of A108540?
%C The ratio A165572(n)/A165571(n) converges towards golden ratio = (1+sqrt(5))/2 = 1.618033988749895... as: 1.5, 1.6666666666666667, 1.5714285714285714, 1.631578947368421, 1.608695652173913, 1.6206896551724137, 1.6185567010309279, 1.6175637393767706, 1.6181172291296626, 1.618066561014263, 1.618063112078346, 1.618031658637302, 1.6180335296782964, 1.6180341824372995, 1.6180339327699054, ...
%H Amiram Eldar, <a href="/A165570/b165570.txt">Table of n, a(n) for n = 1..48</a>
%F a(n) = A165571(n)*A165572(n) = A000040(A165569(n))*A108539(A165569(n)).
%t f[p_] := Module[{x = GoldenRatio * p, p1, p2}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; If[p2 - x > x - p1, p1, p2]]; seq={}; dm = 1; p1 = 1; Do[p1 = NextPrime[p1]; k++; p2 = f[p1]; d = Abs[p2/p1 - GoldenRatio]; If[d < dm, dm = d; AppendTo[seq, p1*p2]], {10^4}]; seq (* _Amiram Eldar_, Nov 28 2019 *)
%Y Cf. A108539, A165569, A165571, A165572.
%K nonn
%O 1,1
%A _Antti Karttunen_, Sep 22 2009
%E a(16)-a(23) from _Donovan Johnson_, May 13 2010
%E a(24)-a(26) from _Amiram Eldar_, Nov 28 2019
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