A164901 a(1)=1, a(2) = 2. For n >=3, a(n) = the smallest integer > a(n-1) that is coprime to every sum of any two distinct earlier terms of this sequence.

1, 2, 4, 7, 13, 19, 29, 37, 43, 53, 59, 67, 73, 79, 89, 97, 103, 109, 127, 137, 149, 157, 163, 173, 179, 191, 197, 211, 223, 229, 239, 251, 257, 263, 269, 277, 283, 293, 307, 313, 331, 337, 347, 353, 359, 367, 373, 379, 389, 397, 409, 419, 431, 439, 449, 457

Offset

1,2

Comments

Except for 1 & 4, all the rest of the terms are prime. They have a density of about 4/5 of the primes for the first 2500 terms. Primes not a member of this sequence: 3, 5, 11, 17, 23, 31, 41, 47, 61, 71, 83, 101, 107, 113, 131, 139, 151, 167, 181, 193, 199, ..., . - Robert G. Wilson v, Aug 31 2009

Conjecture: No two consecutive primes are absent from this sequence (other than 3 and 5). (Equivalently, if p < q < r are consecutive primes and q > 5 is not a term, then p and r are terms.) See A164980. - Rick L. Shepherd, Sep 03 2009

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..3500

Example

The first 4 terms are 1,2,4,7. The sums of every pair of distinct terms are: 1+2=3, 1+4=5, 2+4=6, 7+1=8, 7+2=9, and 7+4=11. So we are looking for the smallest integer > 7 that is coprime to 3, 5, 6, 8, 9, and 11. This number, which is a(5), is 13.

Mathematica

a[1] = 1; a[2] = 2; a[n_] := a[n] = Block[{k = a[n - 1] + 1, t = Plus @@@ Subsets[Array[a, n - 1], {2}]}, While[ Union@ GCD[t, k] != {1}, k++ ]; k]; Array[a, 59] (* Robert G. Wilson v, Aug 31 2009 *)

Crossrefs

Cf. A164902, A164903, A164980.

Sequence in context: A255173 A002466 A162842 * A262744 A359516 A347703

Adjacent sequences: A164898 A164899 A164900 * A164902 A164903 A164904

Keyword

nonn

Author

Leroy Quet, Aug 30 2009

Extensions

a(12) and onward from Robert G. Wilson v, Aug 31 2009

Status

approved