A339943 Let N(p,i) denote the result of applying "nextprime" i times to p; a(n) = smallest prime p such that N(p,3) - p = 2*n, or -1 if no such prime exists.

-1, -1, -1, 3, 7, 17, 23, 43, 79, 107, 109, 113, 197, 199, 317, 509, 523, 773, 1823, 1237, 1319, 3119, 1321, 2473, 2153, 4159, 2477, 6491, 5581, 7351, 9551, 9973, 18803, 18593, 24247, 30559, 31883, 33211, 19603, 66191, 37699, 31393, 83117, 43801, 107351, 107357, 69499, 38461, 130859

Offset: 1

Author

Robert G. Wilson v, Dec 23 2020

Keywords

• sign

Comments

This sequence is the third row of A337767.a(n) > 0 and that there are multiple instances for some k where (p_(k+3) - p_k)/2 - 3 = n.

This sequence only cites the first such occurrence.

n:

4: 3, 5, 11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, 5651, ...,

5: 7, 13, 37, 97, 103, 223, 307, 457, 853, 877, 1087, 1297, ...,

6: 17, 19, 29, 31, 41, 59, 61, 67, 71, 127, 227, 229, ...,

7: 23, 47, 53, 89, 137, 149, 167, 179, 257, 263, 419, 449, ...,

8: 43, 73, 151, 157, 163, 181, 277, 337, 367, 373, 433, 487, ...,

9: 79, 83, 131, 139, 173, 193, 211, 233, 239, 251, 331, 349, ...,

10: 107, 293, 311, 353, 359, 389, 401, 479, 503, 653, 719, 839, ...,

etc.

Examples

			a(4) = 3. This represents the beginning of the run of primes {3, 5, 7, 11}. (11 - 3)/2 = 4 and it is the first prime to do so. Others are 5, 11, 101, 191, etc.;
a(5) = 7. This represents the beginning of the run of primes {7, 11, 13, 17}. (17 - 7)/2 = 5 and it is the first prime to do so. Others are 13, 37, 97, 103, etc.;
a(6) = 17. This represents the beginning of the run of primes {17, 19, 23 & 29}. (29 - 17)/2 = 6 and it is the first prime to do so. Others are 19, 29, 31, 41, etc.
		

Links

• Martin Raab, Table of n, a(n) for n = 1..504 (Terms 1..345 from Robert G. Wilson v)

Crossrefs

Cf. A000230, A001223, A144103, A337767, A339944.

Programs

Mathematica

p = 3; q = 5; r = 7; s = 11; tt[_] := 0; While[p < 250000, d = (s - p)/2; If[ tt[d] == 0, tt[d] = p]; {p, q, r, s} = {q, r, s, NextPrime@ s}]; tt@# & /@ Range@ 75