A118836 - OEIS

%I #10 Nov 22 2024 19:34:33

%S 1,6,55,556,7839,118141,2488800,54871741,1374282325,35786212191,

%T 1182319284628,40234641889543,1409394785418633,53597236487797597,

%U 2091701617809524916,96271871655725943733,4719413412748380767833,240786355921823145103216,13247968989113021361444713

%N Denominators of n-th convergent to continued fraction with semiprime terms.

%C Numerators are A118835. A118835/A118836 converges to semiprime continued fraction constant ~ 4.1636688. The first fractions are 4, 25/6, 229/55, 2315/556, 32639/7839, 491900/118141, 10362539/2488800, 228467758/54871741, 5722056489/1374282325, 149001936472/35786212191, 4922785960065/1182319284628, 167523724578682/40234641889543, 5868253146213935/1409394785418633.

%C These are to semiprimes as A001053 are to natural numbers. See also A105815 Decimal expansion of the semiprime nested radical.

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

%F a(n) = denominator of continued fraction [4; 6, 9, 10, 14, ... A001358(n)]. CONTINUANT transform of A001358.

%e a(1) = 1 = denominator of 4/1.

%e a(2) = 6 = denominator of 25/6 = 4+1/6.

%e a(3) = 55 = denominator of 229/55 = 4+1/(6+1/9).

%e a(4) = 556 = denominator of 2315/556 = 4+1/(6+1/(9+(1/10))).

%t sp = Select[Range[10^3], PrimeOmega[#] == 2 &]; Denominator @ Table[ FromContinuedFraction[ Take[sp, i]], {i, 20}] (* Giovanni Resta, Jun 16 2016 *)

%Y Cf. A001358, A001040, A001053, A105815, A118835.

%K easy,frac,nonn

%O 1,2

%A _Jonathan Vos Post_, May 01 2006

%E Corrected and edited by _Giovanni Resta_, Jun 16 2016