A108540 Golden semiprimes: a(n)=p*q and abs(p*phi-q)<1, where phi = golden ratio = (1+sqrt(5))/2.

6, 15, 77, 187, 589, 851, 1363, 2183, 2747, 7303, 10033, 15229, 16463, 17201, 18511, 27641, 35909, 42869, 45257, 53033, 60409, 83309, 93749, 118969, 124373, 129331, 156433, 201563, 217631, 232327, 237077, 255271, 270349, 283663, 303533, 326423

Offset: 1

Author

Reinhard Zumkeller, Jun 09 2005; revised Jun 13 2005

Keywords

• nonn

Examples

			589 = 19*31 and abs(19*phi - 31) = abs(30,7426... - 31) < 1, therefore 589 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Eric Weisstein's World of Mathematics, Golden Ratio
• Eric Weisstein's World of Mathematics, Semiprime

Crossrefs

Cf. A001622, A001358, A050508.

Programs

Mathematica

f[p_] := Module[{x = GoldenRatio * p}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; q = If[x - p1 < p2 - x, p1, p2]; If[Abs[q - x] < 1, q, 0]]; seq = {}; p=1; Do[p = NextPrime[p]; q = f[p]; If[q > 0, AppendTo[seq, p*q]], {100}]; seq (* Amiram Eldar, Nov 28 2019 *)

Formula

a(n) = A108541(n)*A108542(n) = A000040(k)*A108539(k) for some k.

Extensions

Corrected by T. D. Noe, Oct 25 2006