A257718 -id:A257718 - cp's OEIS frontend

A257716 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.

3, 7, 13, 53, 37, 19, 71, 61, 79, 89, 29, 139, 43, 101, 107, 151, 131, 181, 229, 113, 199, 251, 163, 173, 263, 223, 271, 239, 311, 193, 293, 337, 281, 349, 317, 373, 359, 397, 457, 383, 409, 421, 491, 521, 541, 557, 433, 443, 577, 463, 503, 593, 601, 673, 479, 569, 619, 613

Offset: 1

Robert G. Wilson v, May 05 2015

nonn

Conjecture: The union of this sequence and A257717 is A065091.

			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.

Cf. A031215, A073653, A257717, A257718.

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]

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

A257717 a(n) is the smallest odd-indexed prime not included earlier such that a(n) + a(n-1) + a(n-2) is a prime, beginning with a(1) = 5 and a(2) = 11.

Original entry on oeis.org

5, 11, 31, 17, 23, 67, 41, 59, 73, 47, 103, 83, 97, 127, 149, 157, 137, 167, 283, 191, 179, 109, 211, 227, 313, 233, 197, 331, 241, 257, 379, 347, 307, 367, 389, 277, 353, 401, 439, 269, 509, 499, 419, 449, 571, 431, 487, 563, 461, 547, 523, 587, 599, 661, 607, 761, 631, 677

Offset: 1

Robert G. Wilson v, May 05 2015

nonn

Conjecture: The union of this sequence and A257716 is A065091.

			a(3) = 31 since a(1)+a(2) is 16 and 31, whose index equals 11, is the first odd-indexed prime which meets the criteria. 16 + 7 = 23, a prime, but 7 is the 4th prime and therefore cannot be used.

Cf. A031368, A073653, A257716, A257718.

f[s_List] := Block[{p = s[[-2]] + s[[-1]], q = 17}, While[ !PrimeQ[p + q] || MemberQ[s, q], q = NextPrime[q, 2]]; Append[s, q]]; Nest[f, {5, 11}, 56]

v=[5,11];n=1;while(n<100,p=prime(2*n-1);if(isprime(v[#v]+v[#v-1]+p)&&!vecsearch(vecsort(v),p),v=concat(v,p);n=0);n++);v \\ Derek Orr, May 13 2015

cp's OEIS Frontend

A257716 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.

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