A339188 Highly insulated primes (see Comments for definition).

23, 53, 89, 211, 293, 409, 479, 631, 797, 839, 919, 1039, 1259, 1409, 1471, 1511, 1637, 1709, 1847, 1889, 2039, 2099, 2179, 2503, 2579, 2633, 2777, 2819, 2939, 3011, 3049, 3137, 3229, 3271, 3433, 3499, 3593, 3659, 3709, 3779, 3967, 4111, 4177, 4253, 4327, 4409, 4493, 4621, 4703, 4831

Offset: 1

Author

Abhimanyu Kumar, Nov 27 2020

Keywords

• nonn

Comments

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)).

Examples

			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.
		

Links

• François Marques, Table of n, a(n) for n = 1..10000
• Abhimanyu Kumar and Anuraag Saxena, Insulated primes, arXiv:2011.14210 [math.NT], 2020. See also Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 602-612. See p. 610.

Crossrefs

Cf. A000040, A339148 (insulated primes).

Programs

Mathematica

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 *)

PARI

A339188(n) = { \\ Return the list of the first n highly insulated primes
  my( HighInsulated=List([]), D(p)=min(nextprime(p+1)-p-1, p-precprime(p-1)); );
  my( Dpred_ins=D(7), Pcur_ins=13, Dcur_ins=D(Pcur_ins) );
  local( Dpred=D(Pcur_ins), p=nextprime(Pcur_ins+1), Dp=D(p), Pnext=nextprime(p+1), Dnext=D(Pnext) );
  my(SearchNextInsulated() =
       until(Dp > max(Dpred,Dnext),
         Dpred = Dp; p = Pnext;  Dp = Dnext;
         Pnext = nextprime(p+1); Dnext = D(Pnext);
       );
     \\ At this point p is the first insulated prime > Dcur_ins
    );
  while(#HighInsulated max(Dpred_ins,Dp),
      Dpred_ins = Dcur_ins; Pcur_ins  = p; Dcur_ins  = Dp;
      SearchNextInsulated();
    );
    listput(HighInsulated,Pcur_ins);
  );
  return(HighInsulated);
} \\ François Marques, Dec 01 2020