A259131 -id:A259131 - cp's OEIS frontend

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

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

Offset: 1

Douglas Blumeyer, Mar 14 2022

nonn

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.

			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).
From _Peter Luschny_, Mar 16 2022: (Start)
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)

Union of A002878, A077444, A259131, A267797, etc., minus each sequence's first entry.

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

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