A030168 -id:A030168 - cp's OEIS frontend

A116691 Continued fraction expansion of concatenation of semiprimes (A001358).

0, 2, 7, 1, 1, 2, 4, 26, 1, 4, 1, 7, 2, 2, 1, 1, 6, 1, 3, 1, 8, 1, 7, 6, 1, 1, 1, 22, 3, 18, 2, 1, 24, 11, 1, 2, 1, 5, 7, 1, 2, 1, 3, 20, 1, 1, 5, 2, 70, 1, 1, 2, 2, 1, 1, 1, 1, 2, 5, 1, 2, 1, 1, 6, 2, 153, 1, 2, 1, 10, 13, 4, 1, 4, 1, 3, 1, 1, 3, 2, 2, 6, 3, 1, 4, 1, 7, 1, 1, 6, 1, 1, 14, 2, 1, 2, 2, 2, 2, 2

Offset: 0

Jonathan Vos Post, Mar 15 2006

This is the semiprime analog of A030168: continued fraction expansion of Copeland-Erdős constant (concatenated primes). Decimal expansion of real number formed from concatenation of first 101 digits of A001358, to first 50 terms of continued fraction expansion: 0.469101415212225... = 0 + 1/2+ 1/7+ 1/1+ 1/1+ 1/2+ 1/4+ 1/26+ 1/1+ 1/4+ 1/1+... It seems likely that the real number itself is transcendental.

Cf. A033308 Decimal expansion of Copeland-Erdős constant: concatenate primes.

Extended and edited by Charles R Greathouse IV, Apr 25 2010

A129112 Decimal expansion of constant equal to concatenated semiprimes.

Original entry on oeis.org

4, 6, 9, 1, 0, 1, 4, 1, 5, 2, 1, 2, 2, 2, 5, 2, 6, 3, 3, 3, 4, 3, 5, 3, 8, 3, 9, 4, 6, 4, 9, 5, 1, 5, 5, 5, 7, 5, 8, 6, 2, 6, 5, 6, 9, 7, 4, 7, 7, 8, 2, 8, 5, 8, 6, 8, 7, 9, 1, 9, 3, 9, 4, 9, 5, 1, 0, 6, 1, 1, 1, 1, 1, 5, 1, 1, 8, 1, 1, 9, 1, 2, 1, 1, 2, 2, 1, 2, 3, 1, 2, 9

Offset: 1

Jonathan Vos Post, May 24 2007

Is this, as Copeland and Erdos (1946) showed for the Copeland-Erdos constant, a normal number in base 10? I conjecture that it is, despite the fact that the density of odd semiprimes exceeds the density of even semiprimes. What are the first few digits of the continued fraction of this constant? What are the positions of the first occurrence of n in the continued fraction? What are the incrementally largest terms and at what positions do they occur?
Coincides up to n=15 with concatenation of A046368. - M. F. Hasler, Oct 01 2007
Indeed, a theorem of Copeland & Erdős proves that this constant is 10-normal. - Charles R Greathouse IV, Feb 06 2015

			4.691014152122252633343538394649515557586265...

A. H. Copeland and P. Erdős, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), pp. 857-860.
Eric Weisstein's World of Mathematics, Copeland-Erdos Constant.

Cf. A001358, A019518, A030168, A033308 = decimal expansion of Copeland-Erdos constant: concatenate primes, A033309-A033311, A129808.

Flatten[IntegerDigits/@Select[Range[200],PrimeOmega[#]==2&]] (* Harvey P. Dale, Jan 17 2012 *)

print1(4); for(n=6,129, if(bigomega(n)==2, d=digits(n); for(i=1,#d, print1(", "d[i])))) \\ Charles R Greathouse IV, Feb 06 2015

A308633 Continued fraction for the decimal expansion of the concatenation of the terms of A051699 (distance from n to closest prime).

Original entry on oeis.org

0, 4, 1, 3, 5, 9, 1, 2, 2, 4, 7, 1, 246, 1, 2, 2, 1, 116363868, 3, 1, 1, 1, 3, 4, 282, 1, 1, 1, 2, 1, 8, 2, 1, 1, 1, 1, 7, 10, 7, 1, 2, 1, 6, 2, 1, 2, 7, 2, 11, 1, 3, 1, 4, 1, 4, 1, 3, 5, 9, 1, 1, 1, 3, 3, 1, 3, 2, 1, 5, 3, 3, 1, 32, 1, 1, 15, 3, 1, 1, 11, 9, 1

Offset: 0

Paolo P. Lava, Jun 17 2019

Continued fraction for .2100101012101012101012101232101012321... (see A051699).

Very high value for a(17) = 116363868. This should imply that using the first 16 terms we have a good rational approximation of this decimal expansion: 131256182/624999375 is ok up to the 25th decimal digit.

Cf. A030168, A051699.

Digits:=200: with(numtheory): P:=proc(q) local a,b,n; a:=21;
for n from 2 to q do if isprime(n) then a:=10*a; else
b:=min(nextprime(n)-n,n-prevprime(n)); a:=a*10^length(b)+b; fi; od;
op(convert(evalf(a/10^length(a)),confrac,100)); end: P(200);

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A116691 Continued fraction expansion of concatenation of semiprimes (A001358).

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A129112 Decimal expansion of constant equal to concatenated semiprimes.

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A308633 Continued fraction for the decimal expansion of the concatenation of the terms of A051699 (distance from n to closest prime).

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