A210537 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, ....

3, 5, 9, 11, 15, 21, 23, 29, 33, 35, 39, 45, 51, 53, 59, 65, 71, 75, 81, 89, 93, 99, 101, 105, 113, 119, 123, 131, 135, 141, 143, 149, 155, 159, 161, 165, 171, 179, 185, 189, 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

Offset: 1

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 29 2013

Keywords

• nonn

Comments

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.

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.

Examples

			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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Charles R Greathouse IV, PARI/GP code for computing terms of this sequence

Programs

Mathematica

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 *)

PARI

See Greathouse link.