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A352403 Indices of metallic means that are powers of other metallic means.

%I A352403 #31 Mar 18 2022 21:49:58
%S A352403 4,11,14,29,36,76,82,140,199,234,364,393,478,521,536,756,1030,1364,
%T A352403 1764,2236,2786,3420,3571,3775,4144,4287,4964,5886,6916,8060,8886,
%U A352403 9324,9349,10714,12236,13896,15700,16238,17654,18557,19764,22036,24476,27090,29884
%N A352403 Indices of metallic means that are powers of other metallic means.
%C A352403 Metallic mean k is mm(k) = (k + sqrt(k^2 + 4))/2.
%C A352403 The 4th metallic mean (mm), sometimes called the "copper" mean, is mm(4) = (4 + sqrt(16 + 4))/2 = 4.236... This value is also = 1.618...^3, where 1.618... is the 1st mm, the "golden" one, or (1 + sqrt(1 + 4))/2. This can be shown algebraically.
%C A352403 Any odd power of an mm will give another mm. The odd powers of the 1st mm are:
%C A352403 phi^1 =  1.618..., the  1st mm,
%C A352403 phi^3 =  4.236..., the  4th mm,
%C A352403 phi^5 = 11.090..., the 11th mm,
%C A352403 phi^7 = 29.034..., the 29th mm,
%C A352403 etc.
%C A352403 The indices of these mm's are 1, 4, 11, 29, 76, ... (A002878).
%C A352403 In parallel, the powers of the 2nd mm are:
%C A352403 slv^1 =  2.414..., the  2nd mm,
%C A352403 slv^3 = 14.071..., the 14th mm,
%C A352403 slv^5 = 82.012..., the 82nd mm,
%C A352403 etc.
%C A352403 The indices of these mm's are 2, 14, 82, 478, 2786, ... (A077444).
%C A352403 The indices of the mm's for the 3rd mm are A259131. The 4th mm's are A267797.
%C A352403 Every mm produces such a sequence.
%C A352403 The union of all such sequences (excluding their first terms) is this sequence.
%e A352403 76 is a term since mm(76) = mm(1)^9 is a power of an earlier mean (the golden ratio in this case, and 76 is a Lucas number).
%e A352403 From _Peter Luschny_, Mar 16 2022: (Start)
%e A352403 A representation of the values claimed in the definition is given for the first 8 terms by:
%e A352403 ( 4  +  2*sqrt(5)) / 2 = ((1 + sqrt(5)) / 2)^3.
%e A352403 (11  +  5*sqrt(5)) / 2 = ((1 + sqrt(5)) / 2)^5.
%e A352403 (14  +  5*sqrt(8)) / 2 = ((2 + sqrt(8)) / 2)^3.
%e A352403 (29  + 13*sqrt(5)) / 2 = ((1 + sqrt(5)) / 2)^7.
%e A352403 (36  + 10*sqrt(13))/ 2 = ((3 + sqrt(13))/ 2)^3.
%e A352403 (76  + 34*sqrt(5)) / 2 = ((1 + sqrt(5)) / 2)^9.
%e A352403 (82  + 29*sqrt(8)) / 2 = ((2 + sqrt(8)) / 2)^5.
%e A352403 (140 + 26*sqrt(29))/ 2 = ((5 + sqrt(29))/ 2)^3.
%e A352403 (End)
%t A352403 getMetallicMean[n_] := (n + Power[Power[n, 2] + 4, 1 / 2]) / 2;
%t A352403 getMetallicCompositesUpTo[maxCandidateIndex_] := Module[
%t A352403   {sequence, metallicMeanIndex, metallicMean, oddPower, candidateIndex},
%t A352403   sequence = {};
%t A352403   metallicMeanIndex = 1;
%t A352403   While[
%t A352403     True,
%t A352403     (* skip metallic means already shown to be a power of another *)
%t A352403     If[MemberQ[sequence, metallicMeanIndex], metallicMeanIndex++];
%t A352403     metallicMean = getMetallicMean[metallicMeanIndex];
%t A352403     oddPower = 3;
%t A352403     While[
%t A352403       True,
%t A352403       candidateIndex = Floor[Power[metallicMean, oddPower]];
%t A352403       If[
%t A352403         candidateIndex <= maxCandidateIndex,
%t A352403         AppendTo[sequence, candidateIndex];
%t A352403         oddPower += 2,
%t A352403         Break[]
%t A352403       ]
%t A352403     ];
%t A352403     If[
%t A352403       oddPower == 3,
%t A352403       (* no chance of finding further results below the max, if even the first candidate at this index exceeded it *)
%t A352403       Break[],
%t A352403       metallicMeanIndex++
%t A352403     ];
%t A352403   ];
%t A352403   Sort[sequence]
%t A352403 ];
%t A352403 getMetallicCompositesUpTo[50000]
%Y A352403 Union of A002878, A077444, A259131, A267797, etc., minus each sequence's first entry.
%K A352403 nonn
%O A352403 1,1
%A A352403 _Douglas Blumeyer_, Mar 14 2022