OFFSET
1,1
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)).
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.
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).
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Abhimanyu Kumar, Nov 25 2020
STATUS
approved