A118836 Denominators of n-th convergent to continued fraction with semiprime terms.

1, 6, 55, 556, 7839, 118141, 2488800, 54871741, 1374282325, 35786212191, 1182319284628, 40234641889543, 1409394785418633, 53597236487797597, 2091701617809524916, 96271871655725943733, 4719413412748380767833, 240786355921823145103216, 13247968989113021361444713

Offset: 1

Author

Jonathan Vos Post, May 01 2006

Keywords

• easy
• frac
• nonn
• changed

Comments

Numerators are A118835. A118835/A118836 converges to semiprime continued fraction constant ~ 4.1636688.

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

Examples

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

Links

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

Crossrefs

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

Programs

Mathematica

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

Formula

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

Extensions

Corrected and edited by Giovanni Resta, Jun 16 2016