A339148 -id:A339148 - cp's OEIS frontend

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

Abhimanyu Kumar, Nov 27 2020

nonn

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)).

			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.

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.

Cf. A000040, A339148 (insulated primes).

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 *)

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

A339270 a(n) is the largest m such that there is no prime except prime(n) from prime(n)-m+1 to prime(n)+m.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 4, 1, 2, 3, 1, 2, 4, 5, 1, 2, 3, 1, 2, 3, 4, 6, 3, 1, 2, 1, 2, 4, 3, 4, 1, 2, 1, 2, 5, 3, 4, 5, 1, 2, 1, 2, 1, 2, 11, 3, 1, 2, 4, 1, 2, 5, 5, 5, 1, 2, 3, 1, 2, 10, 3, 1, 2, 4, 5, 6, 1, 2, 4, 6, 5, 5, 3, 4, 6, 3, 4, 8, 1, 2, 1, 2, 3, 4, 6, 3, 1, 2, 4, 7

Offset: 1

Abhimanyu Kumar, Nov 29 2020

nonn

For a prime p, the degree of insulation is formally defined as D(p) = Max_{m=1..oo} U where the set U = {m: A000720(p+m) - A000720(p-m) = 1}.

This sequence is employed in defining insulated primes and highly insulated primes.

Robert Israel, 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.

Cf. A000040, A000720, A339148 (insulated primes), A339188 (highly insulated primes).

Related sequences: A046929.

f:= p -> min(nextprime(p)-p-1, p-prevprime(p)): f(2):= 0:
map(f@ithprime, [$1..100]); # Robert Israel, Dec 24 2020

{0}~Join~Array[Min[NextPrime[# + 1] - # - 1, # - NextPrime[# - 1, -1]] &@ Prime@ # &, 91, 2] (* Michael De Vlieger, Dec 11 2020 *)

D(p)={min(nextprime(p+1)-p-1, p-precprime(p-1))}
forprime(p=2, 1000, print1(D(p), ", "))

cp's OEIS Frontend

A339188 Highly insulated primes (see Comments for definition).

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