A108200 Least positive k such that k*n + 1 is a golden semiprime (A108540).

5, 7, 62, 19, 1, 31, 2, 1254, 1692, 85, 912, 49, 20796, 1, 1234, 627, 50, 846, 4, 860, 28, 456, 4076, 418, 34, 10398, 564, 21, 91250, 617, 6, 47670, 304, 25, 6218, 423, 352018, 2, 6932, 430, 9348, 14, 400, 228, 2874, 2038, 324, 209, 12, 17, 5562, 5199, 25784, 282

Offset: 1

Author

Jason Earls, Jun 15 2005

Keywords

• nonn

Comments

Conjecture: for every n > 4 there exists a number k < n^[n/2] such that k*n + 1 is a golden semiprime, where [] is the floor function.

Examples

			a(3) = 62 because 62*3+1 = 187 = 11*17 and 11*phi-17 = 0.7983... < 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. 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]]]; a[n_] := Module[{k = 1}, While[!goldQ[k * n + 1], k++]; k]; Array[a, 54] (* Amiram Eldar, Nov 29 2019 *)