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%I #35 Apr 19 2022 09:30:02
%S 3,5,9,11,15,21,23,29,33,35,39,45,51,53,59,65,71,75,81,89,93,99,101,
%T 105,113,119,123,131,135,141,143,149,155,159,161,165,171,179,185,189,
%U 191,201,203,213,215,219,233,243,245,249,255,263,269,273,275,281,285,291,309,311,315,323,339,341,345,351,353,365,375,383,389,395,399,413,423,425,429,431,441,453,455,465,471,473,479,491,495,501
%N a(1)=3; for n>1, a(n)>a(n-1) is the minimal for which the set {a(1),a(2),...,a(n)} lacks at least one residue mod 2, 3, ....
%C By the construction, for every N>1, the sequence does not contain a full residue system modulo N. The difference of any two primes greater than 3 in this sequence is a multiple of 6.
%C Conjectures: (1) the sequence contains infinitely many "twins" when such differences equal 6; (2) lim a(n)/prime(n)=1 as n goes to infinity.
%H Charles R Greathouse IV, <a href="/A210537/b210537.txt">Table of n, a(n) for n = 1..10000</a>
%H Charles R Greathouse IV, <a href="/A210537/a210537.gp.txt">PARI/GP code for computing terms of this sequence</a>
%e All terms are odd, so {a(1), ...,} does not contain a complete residue system mod 2. All terms are 0 or 2 mod 3, so the sequence does not contain a complete residue system mod 3.
%t s = {3}; Do[AppendTo[s,2+Last@s]; While[r = 1+Range@Length@s; Max[Length /@ Union /@ (Mod[s,#]& /@ r) - r] == 0, s[[-1]]++], {87}]; s (* _Giovanni Resta_, Jan 29 2013 *)
%o (PARI) See Greathouse link.
%K nonn
%O 1,1
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Jan 29 2013
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