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%I #7 Nov 29 2019 04:01:54
%S 5,2,6,61,52,41,28,13,106,87,66,43,18,393,364,333,300,265,228,189,148,
%T 105,60,13,226,175,122,67,10,463,402,339,274,207,138,67,814,739,662,
%U 583,502,419,334,247,158,67,538,443,346,247,146,43,4494,4387,4278,4167,4054
%N Least positive k such that n^2 + k is a golden semiprime (A108540).
%C Conjecture: for every n > 1 there exists a number k < n^3 such that n^2 + k is a golden semiprime.
%H Amiram Eldar, <a href="/A108399/b108399.txt">Table of n, a(n) for n = 1..10000</a>
%e a(4)=61 because 4^2+61 = 77 = 7*11 and 7*phi-11 = 0.326237... < 1.
%t goldQ[n_] := Module[{f = FactorInteger[n]}, If[Length[f] != 2, False, If[Max[f[[;;,2]]] != 1, False, Abs[f[[2,1]] - f[[1,1]] * GoldenRatio] < 1]]]; a[n_] := Module[{k = 1}, While[!goldQ[n^2 + k], k++]; k]; Array[a, 57] (* _Amiram Eldar_, Nov 29 2019 *)
%Y Cf. A108540.
%K easy,nonn
%O 1,1
%A _Jason Earls_, Jul 03 2005
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