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

1,1

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)).

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.

Example

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.

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 *)

Prog

(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<n,
    until(Dcur_ins > 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

Crossrefs

Cf. A000040, A339148 (insulated primes).

Sequence in context: A128473 A132235 A277993 * A051650 A049438 A078854

Adjacent sequences: A339185 A339186 A339187 * A339189 A339190 A339191

Keyword

nonn

Author

Abhimanyu Kumar, Nov 27 2020

Status

approved