A339148 Insulated primes (see Comments for definition).

7, 13, 23, 37, 53, 67, 89, 103, 113, 131, 139, 157, 173, 181, 193, 211, 233, 277, 293, 337, 359, 389, 409, 421, 449, 479, 491, 509, 547, 577, 607, 631, 653, 691, 709, 751, 761, 797, 811, 823, 839, 863, 887, 919, 953, 983

Offset: 1

Author

Abhimanyu Kumar, Nov 25 2020

Keywords

• nonn

Comments

Let the degree of insulation D(p) for a prime p be defined to be the largest m such that the only prime between p-m and p+m inclusive is p. Then the n-th prime is said to be insulated if and only if D(prime(n)) > D(prime(n+1)) and D(prime(n)) > D(prime(n-1)).

Examples

			For the prime triplet (19,23,29), we have D(19)=2, D(23)=4, and D(29)=1. Hence, 23 is an insulated prime.
		

Links

• Michael De Vlieger, 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.

Crossrefs

Cf. A000040, A000720.

Programs

Mathematica

Select[Prime@ Range[2, 166], And[#2 > #1, #2 > #3] & @@ Map[Min[NextPrime[# + 1] - # - 1, # - NextPrime[# - 1, -1]] &, {NextPrime[# + 1], #, NextPrime[# - 1, -1]}] &] (* Michael De Vlieger, Mar 17 2021 *)

PARI

D(p)={min(nextprime(p+1)-p-1, p-precprime(p-1))}
ok(p)={my(d=D(p)); d>D(nextprime(p+1)) && d>D(precprime(p-1))}
forprime(p=3, 1000, if(ok(p), print1(p, ", "))) \\ Andrew Howroyd, Nov 25 2020

Formula

a(n) ~ 19.18*n^1.093 (heuristically accurate fit for n up to one million).

a(n) ~ c*n^(1+epsilon) (conjectured for some constant c,epsilon as n->oo).