A165569 The indexing sequence for successively better golden semiprimes.

1, 2, 4, 8, 9, 10, 25, 71, 103, 115, 157, 231, 329, 1783, 1835, 4476, 5128, 12462, 16274, 25035, 42174, 72589, 85968, 147666, 613726, 1088825, 1112415, 3125316, 3929736, 5742036, 7639447, 25716100, 32780150, 48132247, 76049401, 100464259, 108803364, 186018939

Offset: 1

Author

Antti Karttunen, Sep 22 2009

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..48

Crossrefs

The corresponding semiprimes are given by A165570(n) = A165571(n)*A165572(n).

Cf. A108539.

Programs

Mathematica

f[p_] := Module[{x = GoldenRatio * p, p1, p2}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; If[p2 - x > x - p1, p1, p2]]; seq={}; k=0; dm = 1; p1 = 1; Do[p1 = NextPrime[p1]; k++; p2 = f[p1]; d = Abs[p2/p1 - GoldenRatio]; If[d < dm, dm = d; AppendTo[seq, k]], {10^4}]; seq (* Amiram Eldar, Nov 28 2019 *)

Formula

a(1)=1, and for n>1, a(n) = first such i>a(n-1) that abs(phi - A108539(i)/A000040(i)) < abs(phi - A108539(a(n-1))/A000040(a(n-1))), where phi = (1+sqrt(5))/2 (Golden ratio).

Extensions

a(16)-a(38) from Amiram Eldar, Nov 28 2019