A118835 Numerators of n-th convergent to continued fraction with semiprime terms.

4, 25, 229, 2315, 32639, 491900, 10362539, 228467758, 5722056489, 149001936472, 4922785960065, 167523724578682, 5868253146213935, 223161143280708212, 8709152841093834203, 400844191833597081550, 19650074552687350830153, 1002554646378888489419353, 55160155625391554268894568

Offset: 1

Author

Jonathan Vos Post, May 01 2006

Keywords

• easy
• frac
• nonn

Comments

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.

These are to semiprimes as A001040 are to natural numbers. See also A105815 Decimal expansion of the semiprime nested radical.

Examples

			a(1) = 4 = numerator of 4/1.
a(2) = 25 = numerator of 25/6 = 4+1/6.
a(3) = 229 = numerator of 229/55 = 4+1/(6+1/9).
a(4) = 2315 = numerator of 2315/556 = 4+1/(6+1/(9+(1/10))).
		

Links

• Eric Weisstein's World of Mathematics, Continued Fraction Constant.

Crossrefs

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

Programs

Mathematica

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

Formula

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

Extensions

Data corrected by Giovanni Resta, Jun 16 2016