A339944 Let N(p,i) denote the result of applying "nextprime" i times to p; a(n) = smallest prime p such that N(p,4) - p = 2*n, or -1 if no such prime exists.

-1, -1, -1, -1, 3, 5, 17, 13, 19, 47, 79, 73, 113, 109, 193, 317, 313, 521, 503, 523, 887, 1499, 1231, 1319, 1373, 1321, 1307, 3947, 2473, 2143, 2477, 7369, 5573, 5939, 9967, 16111, 18587, 20773, 18593, 31883, 17209, 19597, 24251, 19609, 25471, 31397, 44389, 18803, 38459, 38461, 66191, 69557, 103183

Offset: 1

Author

Robert G. Wilson v, Dec 23 2020

Keywords

• sign

Comments

This sequence is the fourth row of A337767.

From Robert G. Wilson v, Dec 30 2020: (Start)

a(n) > -1 for all n >= 5.

It seems likely that for almost all values of n there is more than one integer k such that prime(k+4) - prime(k) = 2*n; a(n) = prime(k) for the smallest such k.

.

n | list of numbers k such that prime(k+4) - prime(k) = 2*n

---+-----------------------------------------------------------------

5 | 3.

6 | 5, 7, 11, 97, 101, 1481, 1867, 3457, 5647, 15727, 16057, ...

7 | 17, 29, 59, 227, 269, 1277, 1289, 1607, 2129, 2789, 3527, ...

8 | 13, 31, 37, 67, 223, 1087, 1291, 1423, 1483, 1597, 1861, ...

9 | 19, 23, 41, 43, 53, 61, 71, 89, 149, 163, 179, ...

10 | 47, 83, 131, 137, 173, 191, 251, 257, 347, 419, 443, ...

etc.

(End)

Examples

			a(1) = 3. This represents the beginning of the run of primes {3, 5, 7, 11, 13}. (13 - 3)/2 = 5 and it is the only prime to do so.
a(2) = 5. This represents the beginning of the run of primes {5, 7, 11, 13, 17}. (17 - 5)/2 = 6 and it is the first prime to do so. Others are 7, 11, 97, 101, etc.
a(3) = 17. This represents the beginning of the run of primes {17, 19, 23, 29, 31}. (31 - 17)/2 = 7 and it is the first prime to do so. Others are 29, 59, 227, 269, etc.
		

Links

• Martin Raab, Table of n, a(n) for n = 1..546 (Terms 1..342 from Robert G. Wilson)

Crossrefs

Cf. A337767, A000230, A144103, A339943.

Programs

Mathematica

p = 3; q = 5; r = 7; s = 11; t = 13; tt[] := 0; While[p < 450000, d = (t - p)/2; If[ tt[d] == 0, tt[d] = p]; {p, q, r, s, t} = {q, r, s, t, NextPrime@ t}]; tt@# & /@ Range@ 75 (* _Robert G. Wilson v, Dec 30 2020 *)