A268343 Hermit primes: primes which are not a nearest neighbor of another prime.

23, 37, 53, 67, 89, 97, 113, 157, 173, 211, 233, 277, 293, 307, 317, 359, 389, 409, 449, 457, 467, 479, 509, 577, 607, 631, 653, 691, 719, 751, 839, 853, 863, 877, 887, 919, 929, 1039, 1069, 1087, 1201, 1223, 1237, 1283, 1297, 1307, 1327, 1381, 1423, 1439

Offset: 1

Author

Karl W. Heuer, Feb 02 2016

Keywords

• easy
• nonn

Comments

If p is a balanced prime (A006562), with two nearest neighbors, then it eliminates both of those neighbors from being hermits.

Conjecture: the asymptotic probability of a prime being in this list is 1/4.

A subsequence of the isolated primes A007510. The sequence of lonely primes A087770 appears to be a subsequence, except for its first three terms (2, 3 and 7). (This would not be true if one of these were followed by two increasingly larger gaps.) - M. F. Hasler, Mar 15 2016

Examples

			53 is in the list because the previous prime, 47, is closer to 43 than to 53, and the following prime, 59, is closer to 61 than to 53.
		

Links

• Karl W. Heuer, Table of n, a(n) for n = 1..30000
• Robert Israel, Table of n, a(n) for n = 1..2600035

Crossrefs

Cf. A269734 (number of hermit primes <= prime(n)).

Programs

Maple

N:= 1000: # to get all terms <= N
pr:= select(isprime, [$2 .. nextprime(nextprime(N))]):
Np:= nops(pr):
ishermit:= Vector(Np,1):
d:= pr[3..Np] + pr[1..Np-2] - 2*pr[2..Np-1]:
for i from 1 to Np-2 do
  if d[i] > 0 then ishermit[i]:= 0
elif d[i] < 0 then ishermit[i+2]:= 0
else ishermit[i]:= 0; ishermit[i+2]:= 0
fi
od:
subs(0=NULL, zip(`*`, pr[1..Np-2],convert(ishermit,list))); # Robert Israel, Mar 09 2016

Mathematica

Select[Prime@ Range@ 228, Function[n, AllTrue[{Subtract @@ Take[#, 2], Subtract @@ Reverse@ Take[#, -2]} &@ Differences[NextPrime[n, #] & /@ {-2, -1, 0, 1, 2}], # < 0 &]]] (* Michael De Vlieger, Feb 02 2016, Version 10 *)

PARI

A268343_list(LIM=1500)={my(d=vector(4),i,o,L=List());forprime(p=1,LIM,(d[i++%4+1]=-o+o=p)d[(i-3)%4+1]&&listput(L,p-d[i%4+1]-d[(i-1)%4+1]));Vec(L)} \\ M. F. Hasler, Mar 15 2016

PARI

is_A268343(n,p=precprime(n-1))={n-p>p-precprime(p-1)&&(p=nextprime(n+1))-n>nextprime(p+1)-p&&isprime(n)} \\ M. F. Hasler, Mar 15 2016

Extensions

Deleted my incorrect conjecture about asymptotic behavior. - N. J. A. Sloane, Mar 10 2016