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A339188 Highly insulated primes (see Comments for definition).

%I A339188 #34 Oct 29 2024 12:21:41
%S A339188 23,53,89,211,293,409,479,631,797,839,919,1039,1259,1409,1471,1511,
%T A339188 1637,1709,1847,1889,2039,2099,2179,2503,2579,2633,2777,2819,2939,
%U A339188 3011,3049,3137,3229,3271,3433,3499,3593,3659,3709,3779,3967,4111,4177,4253,4327,4409,4493,4621,4703,4831
%N A339188 Highly insulated primes (see Comments for definition).
%C A339188 Let degree of insulation D(p) for a prime p be defined as the largest m such that the prime between p-m and p+m is p only. Then the n-th insulated prime is said to be highly insulated if and only if D(A339148(n)) > D(A339148(n+1)) and D(A339148(n)) > D(A339148(n-1)).
%H A339188 François Marques, <a href="/A339188/b339188.txt">Table of n, a(n) for n = 1..10000</a>
%H A339188 Abhimanyu Kumar and Anuraag Saxena, <a href="https://arxiv.org/abs/2011.14210">Insulated primes</a>, arXiv:2011.14210 [math.NT], 2020. See also <a href="https://doi.org/10.7546/nntdm.2024.30.3.602-612">Notes Num. Theor. Disc. Math.</a> (2024) Vol. 30, No. 3, 602-612. See p. 610.
%e A339188 For the triplet (13,23,37) of insulated primes, the values of degree of insulation are D(13)=2, D(23)=4, and D(37)=3. Hence, 23 is the highly insulated prime.
%t A339188 Block[{s = {0}~Join~Array[Min[NextPrime[# + 1] - # - 1, # - NextPrime[# - 1, -1]] &@ Prime@ # &, 660, 2], t}, t = Array[If[#1 < #2 > #3, #4, Nothing] & @@ Append[s[[# - 1 ;; # + 1]], #] &, Length@ s - 2, 2]; Array[If[s[[#1]] < s[[#2]] > s[[#3]], #4, Nothing] & @@ Append[t[[# - 1 ;; # + 1]], Prime@ t[[#]]] &, Length@ t - 2, 2] ] (* _Michael De Vlieger_, Dec 11 2020 *)
%o A339188 (PARI)
%o A339188 A339188(n) = { \\ Return the list of the first n highly insulated primes
%o A339188   my( HighInsulated=List([]), D(p)=min(nextprime(p+1)-p-1, p-precprime(p-1)); );
%o A339188   my( Dpred_ins=D(7), Pcur_ins=13, Dcur_ins=D(Pcur_ins) );
%o A339188   local( Dpred=D(Pcur_ins), p=nextprime(Pcur_ins+1), Dp=D(p), Pnext=nextprime(p+1), Dnext=D(Pnext) );
%o A339188   my(SearchNextInsulated() =
%o A339188        until(Dp > max(Dpred,Dnext),
%o A339188          Dpred = Dp; p = Pnext;  Dp = Dnext;
%o A339188          Pnext = nextprime(p+1); Dnext = D(Pnext);
%o A339188        );
%o A339188      \\ At this point p is the first insulated prime > Dcur_ins
%o A339188     );
%o A339188   while(#HighInsulated<n,
%o A339188     until(Dcur_ins > max(Dpred_ins,Dp),
%o A339188       Dpred_ins = Dcur_ins; Pcur_ins  = p; Dcur_ins  = Dp;
%o A339188       SearchNextInsulated();
%o A339188     );
%o A339188     listput(HighInsulated,Pcur_ins);
%o A339188   );
%o A339188   return(HighInsulated);
%o A339188 } \\ _François Marques_, Dec 01 2020
%Y A339188 Cf. A000040, A339148 (insulated primes).
%K A339188 nonn
%O A339188 1,1
%A A339188 _Abhimanyu Kumar_, Nov 27 2020