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A339188 Highly insulated primes (see Comments for definition). 2
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 (list; graph; refs; listen; history; text; internal format)
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.

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

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Last modified June 24 02:56 EDT 2021. Contains 345415 sequences. (Running on oeis4.)