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

4, 18, 133, 1197, 11374, 110487, 1084605, 10708555, 106091745, 1053422339, 10475688327, 104287176419, 1039019056246, 10358018863853, 103307491450820, 1030734020030318, 10287026204717358, 102692313540015924, 1025351434864118026, 10239531292310798956, 102270102190290407386

Offset: 0

Author

Labos Elemer, Nov 26 2001

Keywords

• nonn
• hard

Examples

			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.
		

References

• A. E. Bojarincev, Asymptotic expressions for the n-th composite number. Univ. Mat. Zap. 6:21-43(1967). [in Russian]
• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 133, p. 45, Ellipses, Paris 2008.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..22
• Laurentiu Panaitopol, Some properties of the series of composed [sic] numbers, Journal of Inequalities in Pure and Applied Mathematics 2:3 (2001).
• J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), pp. 64-94.

Crossrefs

Cf. A033844, A002808, A062298, A000720, A006988, A065855, A065856.

Programs

Mathematica

Composite[n_Integer] := Block[ {k = n + PrimePi[n] + 1 }, While[ k != n + PrimePi[k] + 1, k = n + PrimePi[k] + 1]; Return[k]];
Table[Composite[10^n], {n, 0, 9}]

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

Formula

a(n) = A002808(A011557(n)).

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

Extensions

More terms from Robert G. Wilson v, Nov 26 2001

a(14) from Lekraj Beedassy, Jul 14 2008

a(15)-a(19) from Chai Wah Wu, Apr 16 2018

a(20) from Chai Wah Wu, Aug 23 2018