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A352403
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Indices of metallic means that are powers of other metallic means.
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0
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4, 11, 14, 29, 36, 76, 82, 140, 199, 234, 364, 393, 478, 521, 536, 756, 1030, 1364, 1764, 2236, 2786, 3420, 3571, 3775, 4144, 4287, 4964, 5886, 6916, 8060, 8886, 9324, 9349, 10714, 12236, 13896, 15700, 16238, 17654, 18557, 19764, 22036, 24476, 27090, 29884
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OFFSET
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1,1
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COMMENTS
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Metallic mean k is mm(k) = (k + sqrt(k^2 + 4))/2.
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.
Any odd power of an mm will give another mm. The odd powers of the 1st mm are:
phi^1 = 1.618..., the 1st mm,
phi^3 = 4.236..., the 4th mm,
phi^5 = 11.090..., the 11th mm,
phi^7 = 29.034..., the 29th mm,
etc.
The indices of these mm's are 1, 4, 11, 29, 76, ... (A002878).
In parallel, the powers of the 2nd mm are:
slv^1 = 2.414..., the 2nd mm,
slv^3 = 14.071..., the 14th mm,
slv^5 = 82.012..., the 82nd mm,
etc.
The indices of these mm's are 2, 14, 82, 478, 2786, ... (A077444).
The indices of the mm's for the 3rd mm are A259131. The 4th mm's are A267797.
Every mm produces such a sequence.
The union of all such sequences (excluding their first terms) is this sequence.
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LINKS
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EXAMPLE
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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).
A representation of the values claimed in the definition is given for the first 8 terms by:
( 4 + 2*sqrt(5)) / 2 = ((1 + sqrt(5)) / 2)^3.
(11 + 5*sqrt(5)) / 2 = ((1 + sqrt(5)) / 2)^5.
(14 + 5*sqrt(8)) / 2 = ((2 + sqrt(8)) / 2)^3.
(29 + 13*sqrt(5)) / 2 = ((1 + sqrt(5)) / 2)^7.
(36 + 10*sqrt(13))/ 2 = ((3 + sqrt(13))/ 2)^3.
(76 + 34*sqrt(5)) / 2 = ((1 + sqrt(5)) / 2)^9.
(82 + 29*sqrt(8)) / 2 = ((2 + sqrt(8)) / 2)^5.
(140 + 26*sqrt(29))/ 2 = ((5 + sqrt(29))/ 2)^3.
(End)
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MATHEMATICA
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getMetallicMean[n_] := (n + Power[Power[n, 2] + 4, 1 / 2]) / 2;
getMetallicCompositesUpTo[maxCandidateIndex_] := Module[
{sequence, metallicMeanIndex, metallicMean, oddPower, candidateIndex},
sequence = {};
metallicMeanIndex = 1;
While[
True,
(* skip metallic means already shown to be a power of another *)
If[MemberQ[sequence, metallicMeanIndex], metallicMeanIndex++];
metallicMean = getMetallicMean[metallicMeanIndex];
oddPower = 3;
While[
True,
candidateIndex = Floor[Power[metallicMean, oddPower]];
If[
candidateIndex <= maxCandidateIndex,
AppendTo[sequence, candidateIndex];
oddPower += 2,
Break[]
]
];
If[
oddPower == 3,
(* no chance of finding further results below the max, if even the first candidate at this index exceeded it *)
Break[],
metallicMeanIndex++
];
];
Sort[sequence]
];
getMetallicCompositesUpTo[50000]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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