cp's OEIS Frontend

A065857 The (10^n)-th composite number.

%I A065857 #51 Mar 08 2020 17:15:34
%S A065857 4,18,133,1197,11374,110487,1084605,10708555,106091745,1053422339,
%T A065857 10475688327,104287176419,1039019056246,10358018863853,
%U A065857 103307491450820,1030734020030318,10287026204717358,102692313540015924,1025351434864118026,10239531292310798956,102270102190290407386
%N A065857 The (10^n)-th composite number.
%D A065857 A. E. Bojarincev, Asymptotic expressions for the n-th composite number. Univ. Mat. Zap. 6:21-43(1967). [in Russian]
%D A065857 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 133, p. 45, Ellipses, Paris 2008.
%H A065857 Chai Wah Wu, <a href="/A065857/b065857.txt">Table of n, a(n) for n = 0..22</a>
%H A065857 Laurentiu Panaitopol, <a href="https://www.emis.de/journals/JIPAM/article153.html?sid=153">Some properties of the series of composed [sic] numbers</a>, Journal of Inequalities in Pure and Applied Mathematics 2:3 (2001).
%H A065857 J. Barkley Rosser and Lowell Schoenfeld, <a href="http://projecteuclid.org/euclid.ijm/1255631807">Approximate formulas for some functions of prime numbers</a>. Illinois J. Math. 6 (1962), pp. 64-94.
%F A065857 a(n) = A002808(A011557(n)).
%F A065857 a(n) = 10^(n + n/log n + 2n/log^2 + 4n/log^3 n + O(n/log^4 n)). See Bojarincev for an asymptotic expansion. - _Charles R Greathouse IV_, May 30 2013
%e A065857 The 100th composite number is C(100)=133, while the 100th prime is 541. In general: A000720(m) < A062298(m) < m < A002808(m) < A000040(m), for example pi(100)=25 < 75 < 100 < C(100)=133 < prime(100)=541.
%t A065857 Composite[n_Integer] := Block[ {k = n + PrimePi[n] + 1 }, While[ k != n + PrimePi[k] + 1, k = n + PrimePi[k] + 1]; Return[k]];
%t A065857 Table[Composite[10^n], {n, 0, 9}]
%o A065857 (PARI) a(n)=my(k=10^n);forcomposite(n=4,2*k+2,if(k--==0,return(n))) \\ _Charles R Greathouse IV_, May 30 2013
%Y A065857 Cf. A033844, A002808, A062298, A000720, A006988, A065855, A065856.
%K A065857 nonn,hard
%O A065857 0,1
%A A065857 _Labos Elemer_, Nov 26 2001
%E A065857 More terms from _Robert G. Wilson v_, Nov 26 2001
%E A065857 a(14) from _Lekraj Beedassy_, Jul 14 2008
%E A065857 a(15)-a(19) from _Chai Wah Wu_, Apr 16 2018
%E A065857 a(20) from _Chai Wah Wu_, Aug 23 2018