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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.

%I A339944 #48 Feb 10 2024 13:34:40
%S A339944 -1,-1,-1,-1,3,5,17,13,19,47,79,73,113,109,193,317,313,521,503,523,
%T A339944 887,1499,1231,1319,1373,1321,1307,3947,2473,2143,2477,7369,5573,5939,
%U A339944 9967,16111,18587,20773,18593,31883,17209,19597,24251,19609,25471,31397,44389,18803,38459,38461,66191,69557,103183
%N 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.
%C A339944 This sequence is the fourth row of A337767.
%C A339944 From _Robert G. Wilson v_, Dec 30 2020: (Start)
%C A339944 a(n) > -1 for all n >= 5.
%C A339944 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.
%C A339944 .
%C A339944    n | list of numbers k such that prime(k+4) - prime(k) = 2*n
%C A339944   ---+-----------------------------------------------------------------
%C A339944    5 |  3.
%C A339944    6 |  5,  7,  11,  97, 101, 1481, 1867, 3457, 5647, 15727, 16057, ...
%C A339944    7 | 17, 29,  59, 227, 269, 1277, 1289, 1607, 2129,  2789,  3527, ...
%C A339944    8 | 13, 31,  37,  67, 223, 1087, 1291, 1423, 1483,  1597,  1861, ...
%C A339944    9 | 19, 23,  41,  43,  53,   61,   71,   89,  149,   163,   179, ...
%C A339944   10 | 47, 83, 131, 137, 173,  191,  251,  257,  347,   419,   443, ...
%C A339944 etc.
%C A339944 (End)
%H A339944 Martin Raab, <a href="/A339944/b339944.txt">Table of n, a(n) for n = 1..546</a> (Terms 1..342 from Robert G. Wilson)
%e A339944 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.
%e A339944 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.
%e A339944 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.
%t A339944 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 *)
%Y A339944 Cf. A337767, A000230, A144103, A339943.
%K A339944 sign
%O A339944 1,5
%A A339944 _Robert G. Wilson v_, Dec 23 2020