A038179 - OEIS

%I #89 Oct 26 2025 14:59:40

%S 2,3,5,7,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,55,59,61,65,67,

%T 71,73,77,79,83,85,89,91,95,97,101,103,107,109,113,115,119,121,125,

%U 127,131,133,137,139,143,145,149,151,155,157,161,163,167,169,173,175

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

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

%C Apart from first two terms, same as A007310.

%C 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

%C 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

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

%H G. C. Greubel, <a href="/A038179/b038179.txt">Table of n, a(n) for n = 1..2500</a>

%H Ahmed Hamdy A. Diab, <a href="https://arxiv.org/abs/2012.03052">Sequence eliminating law (SEL) and the interval formulas of prime numbers</a>, arXiv:2012.03052 [math.NT], 2020.

%H H. B. Meyer, <a href="http://www.hbmeyer.de/eratosiv.htm">Eratosthenes' sieve</a>.

%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F O.g.f.: x*(2 + x + x^3 + 2x^4)/((1+x)*(1-x)^2). - _R. J. Mathar_, May 23 2008

%F a(n) = (1/9)*(4*n^3 + 3*n^2 + 1 - Kronecker(-3,n+1)). - _Ralf Stephan_, Jun 01 2014

%F From _Mikk Heidemaa_, Oct 28 2017: (Start)

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

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

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

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

%t max = 200; Complement[Range[2, max], 2Range[2, Ceiling[max/2]], 6Range[2, Ceiling[max/6]] + 3] (* _Alonso del Arte_, May 16 2014 *)

%t Prepend[Table[3*n - Mod[ Mod[n, 2] + 1, n], {n, 1, 999}], 2] (* _Mikk Heidemaa_, Nov 02 2017 *)

%Y Cf. A004280, A007310, A144065.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, Dec 11 1999

%E Name edited by _Michel Marcus_, Dec 14 2020