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%I #17 Mar 02 2018 02:11:56
%S 1,2,4,7,13,19,29,37,43,53,59,67,73,79,89,97,103,109,127,137,149,157,
%T 163,173,179,191,197,211,223,229,239,251,257,263,269,277,283,293,307,
%U 313,331,337,347,353,359,367,373,379,389,397,409,419,431,439,449,457
%N 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.
%C 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
%C 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
%H Robert G. Wilson v, <a href="/A164901/b164901.txt">Table of n, a(n) for n = 1..3500</a>
%e 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.
%t 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 *)
%Y Cf. A164902, A164903, A164980.
%K nonn
%O 1,2
%A _Leroy Quet_, Aug 30 2009
%E a(12) and onward from _Robert G. Wilson v_, Aug 31 2009
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