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

0,1

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.

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

Example

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.

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}]

Prog

(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

Crossrefs

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

Sequence in context: A194559 A356542 A371038 * A214647 A156445 A304997

Adjacent sequences: A065854 A065855 A065856 * A065858 A065859 A065860

Keyword

nonn,hard

Author

Labos Elemer, Nov 26 2001

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

Status

approved