A118835 - OEIS

%I #8 Jun 16 2016 22:23:00

%S 4,25,229,2315,32639,491900,10362539,228467758,5722056489,

%T 149001936472,4922785960065,167523724578682,5868253146213935,

%U 223161143280708212,8709152841093834203,400844191833597081550,19650074552687350830153,1002554646378888489419353

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

%C Denominators are A118836. 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 A001040 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) = numerator of continued fraction [4; 6, 9, 10, 14, ... A001358(n)]. CONTINUANT transform of A001358.

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

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

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

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

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

%Y Cf. A001358, A001040, A001053, A105815, A118836.

%K easy,frac,nonn

%O 1,1

%A _Jonathan Vos Post_, May 01 2006

%E Data corrected by _Giovanni Resta_, Jun 16 2016