A337767 -id:A337767 - cp's OEIS frontend

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

Robert G. Wilson v, Dec 23 2020

sign

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.

			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.

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

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

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

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.

Original entry on oeis.org

-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

Robert G. Wilson v, Dec 23 2020

sign

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)

			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.

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

Cf. A337767, A000230, A144103, A339943.

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 *)

cp's OEIS Frontend

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.

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