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a(2n-1) is the first unused prime which is the sum of the two preceding terms and such that a(2n) is that sum and it is also an unused prime.
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%I #17 Jan 03 2013 16:02:46

%S 3,5,11,19,7,37,17,61,23,101,13,137,29,179,31,239,41,311,67,419,71,

%T 557,73,701,47,821,43,911,59,1013,79,1151,53,1283,97,1433,83,1613,127,

%U 1823,89,2039,109,2237,113,2459,139,2711,103,2953,107,3163,163,3433,131,3727,149,4007,181,4337,173,4691

%N a(2n-1) is the first unused prime which is the sum of the two preceding terms and such that a(2n) is that sum and it is also an unused prime.

%H Vincenzo Librandi, <a href="/A198519/b198519.txt">Table of n, a(n) for n = 1..10000</a>

%e a(3) does not equal 7 since 3+5+7 which is 15 is not a prime, but the next prime, 11, meets the criteria.

%t s = {3, 5}; k = 1; While[k < 31, p = s[[-2]] + s[[-1]]; q = 7; While[ !PrimeQ[p + q] || MemberQ[s, q] || MemberQ[s, p + q], q = NextPrime@ q]; AppendTo[s, q]; AppendTo[s, p + q]; k++]; s

%Y Cf. A073653, A084331, A101595.

%K easy,nonn

%O 1,1

%A _Robert G. Wilson v_, Dec 21 2012