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%I #16 Mar 23 2024 12:19:48
%S 3,7,13,53,37,19,71,61,79,89,29,139,43,101,107,151,131,181,229,113,
%T 199,251,163,173,263,223,271,239,311,193,293,337,281,349,317,373,359,
%U 397,457,383,409,421,491,521,541,557,433,443,577,463,503,593,601,673,479,569,619,613
%N a(n) = smallest prime of even index not included earlier such that a(n) + a(n-1) + a(n-2) is a prime, beginning with a(1) = 3 and a(2) = 7.
%C Conjecture: The union of this sequence and A257717 is A065091.
%e a(4) = 53 since a(2)+a(3) is 20 and 53, whose index equals 16, is the first even-indexed prime which meets the criteria. 20 + 11 = 31, a prime, but 11 is the 5th prime and therefore cannot be used.
%t f[s_List] := Block[{p = s[[-2]] + s[[-1]], q = 13}, While[ !PrimeQ[p + q] || MemberQ[s, q], q = NextPrime[q, 2]]; Append[s, q]]; Nest[f, {3, 7}, 56]
%o (PARI) v=[3,7];n=1;while(n<100,if(isprime(v[#v]+v[#v-1]+prime(2*n))&&!vecsearch(vecsort(v),prime(2*n)),v=concat(v,prime(2*n));n=0);n++);v \\ _Derek Orr_, May 13 2015
%Y Cf. A031215, A073653, A257717, A257718.
%K nonn
%O 1,1
%A _Robert G. Wilson v_, May 05 2015
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