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A108540 Golden semiprimes: a(n)=p*q and abs(p*phi-q)<1, where phi = golden ratio = (1+sqrt(5))/2. 17

%I

%S 6,15,77,187,589,851,1363,2183,2747,7303,10033,15229,16463,17201,

%T 18511,27641,35909,42869,45257,53033,60409,83309,93749,118969,124373,

%U 129331,156433,201563,217631,232327,237077,255271,270349,283663,303533,326423

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

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

%D Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.

%H Amiram Eldar, <a href="/A108540/b108540.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>

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

%t 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 *)

%Y Cf. A001358, A050508.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Jun 09 2005; revised Jun 13 2005

%E Corrected by _T. D. Noe_, Oct 25 2006

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Last modified June 16 19:05 EDT 2021. Contains 345068 sequences. (Running on oeis4.)