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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.

%I A339943 #32 Feb 10 2024 13:35:05
%S A339943 -1,-1,-1,3,7,17,23,43,79,107,109,113,197,199,317,509,523,773,1823,
%T A339943 1237,1319,3119,1321,2473,2153,4159,2477,6491,5581,7351,9551,9973,
%U A339943 18803,18593,24247,30559,31883,33211,19603,66191,37699,31393,83117,43801,107351,107357,69499,38461,130859
%N 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.
%C A339943 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.
%C A339943 This sequence only cites the first such occurrence.
%C A339943 n:
%C A339943 4:    3,   5,  11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, 5651, ...,
%C A339943 5:    7,  13,  37,  97, 103, 223,  307,  457,  853,  877, 1087, 1297, ...,
%C A339943 6:   17,  19,  29,  31,  41,  59,   61,   67,   71,  127,  227,  229, ...,
%C A339943 7:   23,  47,  53,  89, 137, 149,  167,  179,  257,  263,  419,  449, ...,
%C A339943 8:   43,  73, 151, 157, 163, 181,  277,  337,  367,  373,  433,  487, ...,
%C A339943 9:   79,  83, 131, 139, 173, 193,  211,  233,  239,  251,  331,  349, ...,
%C A339943 10: 107, 293, 311, 353, 359, 389,  401,  479,  503,  653,  719,  839, ...,
%C A339943 etc.
%H A339943 Martin Raab, <a href="/A339943/b339943.txt">Table of n, a(n) for n = 1..504</a> (Terms 1..345 from Robert G. Wilson v)
%e A339943 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.;
%e A339943 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.;
%e A339943 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.
%t A339943 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
%Y A339943 Cf. A000230, A001223, A144103, A337767, A339944.
%K A339943 sign
%O A339943 1,4
%A A339943 _Robert G. Wilson v_, Dec 23 2020