A108219 - OEIS

%I #11 Nov 29 2019 04:01:58

%S 8,9,26,44,105,112,125,126,150,160,180,192,216,243,292,568,639,1174,

%T 1407,1448,1629,1675,2010,2144,2379,2412,2685,2722,2864,3222,3355,

%U 3835,3999,4026,4107,4543,4602,5035,5709,5978,6042,6235,6307,6355,6490,7482

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

%C 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.

%H Amiram Eldar, <a href="/A108219/b108219.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 *)

%Y Cf. A001414, A108540.

%K nonn

%O 1,1

%A _Jason Earls_, Jun 16 2005