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.

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

1,4

Comments

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.

Links

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.

Maple

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

Mathematica

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

Prog

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

Crossrefs

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

Related sequences: A046929.

Sequence in context: A135990 A347256 A347224 * A345226 A140715 A330690

Adjacent sequences: A339267 A339268 A339269 * A339271 A339272 A339273

Keyword

nonn

Author

Abhimanyu Kumar, Nov 29 2020

Status

approved