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

1,1

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.

Links

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

Example

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

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 *)

Crossrefs

Cf. A001414, A108540.

Sequence in context: A305828 A322652 A041134 * A041317 A305336 A042311

Adjacent sequences: A108216 A108217 A108218 * A108220 A108221 A108222

Keyword

nonn

Author

Jason Earls, Jun 16 2005

Status

approved