A118836 -id:A118836 - cp's OEIS frontend

A121661 Duplicate of A118836.

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

Offset: 1

Jonathan Vos Post, Aug 14 2006

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Previous name was: Denominator of fraction equal to the continued fraction [4, 6, 9, ..., semiprime(n)].

Numerator of fraction equal to the continued fraction [4, 6, 9, ..., semiprime(n)] is A121660. Hence A121660/A121661 is semiprime analog of A036247/A036248.

			a(1) = denominator of 4 = 1.
a(2) = denominator of 4 + 1/6 = denominator of 25/6 = 6.
a(3) = denominator of 4 + 1/(6+1/9) = denominator of 229/55 = 55.
a(10) = denominator of 4+1/(6+1/(9+1/(10+ 1/(14+1/(15+ 1/(21+1/(22+1/(25+1/(26))))))))) = denominator of 149001936472/35786212191 = 35786212191.

A121661 := proc(n) local x,i ; x := A001358(n) ; for i from n-1 to 1 by -1 do x := A001358(i)+1/x ; end do: denom(%) ; end proc: # R. J. Mathar, Jun 29 2011

a(n) = A118836(n). - Georg Fischer, Nov 02 2018

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

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, May 01 2006

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.

			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))).

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

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

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

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

Data corrected by Giovanni Resta, Jun 16 2016

cp's OEIS Frontend

A121661 Duplicate of A118836.

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

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