A108219 Numbers n such that A001414(n) is a golden semiprime, where A001414 is the sum of primes dividing n (with repetition).

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

Offset: 1

Author

Jason Earls, Jun 16 2005

Keywords

• nonn

Comments

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.

Examples

			5709 = 3*11*173 is in the sequence because 3+11+173 = 187 = 11*17 and 11*phi-17 = 0.79837... < 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414, A108540.

Programs

Mathematica

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 *)