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A108219
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Numbers n such that A001414(n) is a golden semiprime, where A001414 is the sum of primes dividing n (with repetition).
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1
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8, 9, 26, 44, 105, 112, 125, 126, 150, 160, 180, 192, 216, 243, 292, 568, 639, 1174, 1407, 1448, 1629, 1675, 2010, 2144, 2379, 2412, 2685, 2722, 2864, 3222, 3355, 3835, 3999, 4026, 4107, 4543, 4602, 5035, 5709, 5978, 6042, 6235, 6307, 6355, 6490, 7482
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OFFSET
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1,1
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COMMENTS
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Numbers n such that A001414(n) and A001414(n+1) are both golden semiprimes: 8, 125, 153759, 247455, 678807, 1243499, 1243500, ... Notice that the last two terms indicate a triple. Conjecture: this subsequence is infinite.
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LINKS
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EXAMPLE
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5709 = 3*11*173 is in the sequence because 3+11+173 = 187 = 11*17 and 11*phi-17 = 0.79837... < 1.
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MATHEMATICA
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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]]]; sumPrimes[n_] := Plus @@ Times @@@ FactorInteger[n]; Select[Range[7500], goldQ[sumPrimes[#]] &] (* Amiram Eldar, Nov 29 2019 *)
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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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