A038179 Result of second stage of sieve of Eratosthenes (after eliminating multiples of 2 and 3).

2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151

Offset: 1

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• easy

Comments

2, 3 and numbers of the form 6m +- 1.

Apart from first two terms, same as A007310.

Terms of this sequence (starting from the third term) are equal to the result of the expression sqrt(4!*(k+1) + 1) - but only when this expression yields integral values (that is when the parameter k takes values, which are terms of A144065). - Alexander R. Povolotsky, Sep 09 2008

For every integer n>2, n is in this sequence iff Product_{k=2..oo} 1/(1 - 1/k^n) = Product_{k=1..n} Gamma( 2 - (-1)^(k*(1 + 1/n)) ). - Federico Provvedi, Nov 07 2024

References

• Fred S. Roberts, Applied Combinatorics, Prentice-Hall, 1984, p. 256.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2500
• Ahmed Hamdy A. Diab, Sequence eliminating law (SEL) and the interval formulas of prime numbers, arXiv:2012.03052 [math.NT], 2020.
• H. B. Meyer, Eratosthenes' sieve.
• Index entries for sequences generated by sieves.

Crossrefs

Cf. A004280, A007310, A144065.

Programs

Mathematica

max = 200; Complement[Range[2, max], 2Range[2, Ceiling[max/2]], 6Range[2, Ceiling[max/6]] + 3] (* Alonso del Arte, May 16 2014 *)
Prepend[Table[3*n - Mod[ Mod[n, 2] + 1, n], {n, 1, 999}], 2] (* Mikk Heidemaa, Nov 02 2017 *)

Formula

O.g.f.: x*(2 + x + x^3 + 2x^4)/((1+x)*(1-x)^2). - R. J. Mathar, May 23 2008

a(n) = (1/9)*(4*n^3 + 3*n^2 + 1 - Kronecker(-3,n+1)). - Ralf Stephan, Jun 01 2014

From Mikk Heidemaa, Oct 28 2017: (Start)

a(n) = floor((41/21 - (3 mod n))^(-3*n+5)) + 3*n - 4 (n > 0).

a(n+1) = 3*n - ((n mod 2)+1) mod n (n > 0). (End)

a(n+2) = 2*floor((3*n+1)/2) + 1 for n>=1; see (17) in Diab link. - Michel Marcus, Dec 14 2020

Sum_{n>=1} (-1)^(n+1)/a(n) = (7-sqrt(3)*Pi)/6. - Amiram Eldar, Sep 22 2022

Extensions

Name edited by Michel Marcus, Dec 14 2020