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

1,1

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

Links

Karl W. Heuer, Table of n, a(n) for n = 1..30000

Robert Israel, Table of n, a(n) for n = 1..2600035

Example

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.

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 *)

Prog

(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-1)%4+1]&&d[(i-2)%4+1]>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

Crossrefs

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

Sequence in context: A153740 A055114 A329262 * A063643 A361530 A057876

Adjacent sequences: A268340 A268341 A268342 * A268344 A268345 A268346

Keyword

easy,nonn

Author

Karl W. Heuer, Feb 02 2016

Extensions

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

Status

approved