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a[1]=1; a[2]=2; a[n_]:=a[n]=(k=1; While[Length@Select[Plus@@@Subsets[{a[n-1], a[n-2], ++k}, {2}], PrimeQ]!=2||MemberQ[Array[a, n-1], k]]; k); Array[a, 100] (* Giorgos Kalogeropoulos, May 09 2021 *)

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(PARI) A329411(n, show=0, o=1, N=2, M=2, p=[], U, u=o)={for(n=o, n-1, if(, show>0, && print1(o", "), ", "); show<0, && listput(L, o)); ); U+=1<<(o-u); U>>=-u+u+=valuation(U+1, 2); p=concat(if(#p>=M, , p[^1], ], p), ), o); my(c=N-sum(i=2, #p, , sum(j=1, i-1, isprime(p[i]+p[j])))); if(#p<M&&sum(i=1, #p, isprime(p[i]+u))<=c, o=u)|| for(k=u, oo, bittest(U, k-u)|| summin(i=1, #c-#[0|p<-p, , isprime(p[i]++k))!=c||[)], #p>=M) ||[o=k, break])); show&&print([u]); o} \\ Optional args: show=1: print a(o..n-1), show=-1: append a(o..n-1) to the (global) list L, in both cases print [least unused number] at the end; o=0: start with a(o)=o; N, M: find N primes using M+1 consecutive terms. - M. F. Hasler, Nov 16 2019

Cf. A055265 (sum of two consecutive terms is always prime: differs from a(30) on).

Cf. A055265 (sum of two consecutive terms is always prime: differs from a(30) on), A128280 (same for nonnegative terms).

See also "nonnegative" variants: A253074, A329450 (0 primes using 2 resp. 3 terms), A128280 (1 prime from 2 terms), A329452, A329453 (2 primes usingfrom 4 resp. 5 terms), A329454, A329455 (3 primes usingfrom 4 resp. 5 terms), A329449, A329456 (4 primes from using4 resp. 5 terms).). See the Wiki page for more.

Cf. A329333 (3 consecutive terms, exactly 1 prime sum).

Cf. A329405 (no prime among the pairwise sums of 3 consecutive terms).

Cf. A329406 .. A329410 (exactly 1 prime sum using 4, ..., 10 consecutive terms).

Cf. A055265 (sum of two consecutive terms is always prime: differs from a(30) on), A128280 (same for nonnegative terms).

Cf. A329333, A329406 .. A329410 (exactly 1 prime sum using 3, 4, ..., 10 consecutive terms).

Cf. A055266 (no prime sum among 2 consecutive terms), A329405 (no prime among the pairwise sums of 3 consecutive terms).

See also "nonnegative" variants: A253074, A329450 (0 primes using 2 resp. 3 terms), A329452, A329453 (2 primes using 4 terms), A329453 (2 primes using resp. 5 terms), A329454, A329455 (3 primes using 4 terms), A329449 (4 primes using 4 terms), A329455 (3 primes using resp. 5 terms), A329449, A329456 (4 primes using resp. 5 terms).

Could be extended to a(0) = 0 to yield a sequence of nonnegative integers with the same property, including lexicographic minimality, which is a permutation of the nonnegative integers iff this sequence is a permutation of the positive integers.

Could be extended to a(0) = 0, yielding the sequence of nonnegative integers with the same property (including lexicographic minimality), which is a permutation of the nonnegative integers if this sequence is a permutation of the positive integers. This is the first known example amongwhere the restriction of S(N,M;0) to [1..oo) gives sequencesS(N,M;1), where S(N,M;o) = ) is the lexicographically firstsmallest sequence startingwith a(o)=o with , N primes among pairwise sums of M consecutive terms, where theand restrictionno ofduplicate S(N,M;0) to [1..oo) givesterms: For S(N,M;1): e.g., example, S(0,3;1) = A329405 is not A329450\{0}, S(2,4;1) = A329412 is not A329452\{0}, etc. The second such example is S(4,4;o) = A329449. - M. F. Hasler, Nov 16 2019

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M. F. Hasler: Minor edits to make the comments hopefully easier to read.

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Lexicographically earliest sequence of distinct positive numbers such that amongAmong the pairwise sums of any three consecutive terms there are exactly two prime sums: lexicographically earliest such sequence of distinct positive numbers.

Cf. A329405 (no prime among the pairwise sums of 3 consecutive terms).

Cf. A329406 .. A329410 (exactly 1 prime sum using 4, ..., 10 consecutive terms).

Cf. A329412 .. A329416 (exactly 2 prime sums using 4, ..., 10 consecutive terms).

Cf. See also "nonnegative" variants: A329450 (0 primes using 3 terms), A329452 (2 primes using 4 terms), A329453 (2 primes using 5 terms), A329454 (3 primes using 4 terms), A329449 (4 primes using 4 terms), A329455 (3 primes using 5 terms), A329456 (4 primes using 5 terms).

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M. F. Hasler: Putting the relevant information to the front of NAME, since it was impossible to distinguish all the neighboring sequences from the "popup" titles truncated to 1 line.