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

Offset

1,1

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.

Links

Table of n, a(n) for n=1..18.

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

Formula

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

Example

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))).

Mathematica

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

Crossrefs

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

Sequence in context: A010845 A087660 A121660 * A365053 A224080 A194569

Adjacent sequences: A118832 A118833 A118834 * A118836 A118837 A118838

Keyword

easy,frac,nonn

Author

Jonathan Vos Post, May 01 2006

Extensions

Data corrected by Giovanni Resta, Jun 16 2016

Status

approved