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

%I A268343 #35 Jan 06 2020 18:12:42
%S A268343 23,37,53,67,89,97,113,157,173,211,233,277,293,307,317,359,389,409,
%T A268343 449,457,467,479,509,577,607,631,653,691,719,751,839,853,863,877,887,
%U A268343 919,929,1039,1069,1087,1201,1223,1237,1283,1297,1307,1327,1381,1423,1439
%N A268343 Hermit primes: primes which are not a nearest neighbor of another prime.
%C A268343 If p is a balanced prime (A006562), with two nearest neighbors, then it eliminates both of those neighbors from being hermits.
%C A268343 Conjecture: the asymptotic probability of a prime being in this list is 1/4.
%C A268343 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
%H A268343 Karl W. Heuer, <a href="/A268343/b268343.txt">Table of n, a(n) for n = 1..30000</a>
%H A268343 Robert Israel, <a href="/A268343/a268343.txt">Table of n, a(n) for n = 1..2600035</a>
%e A268343 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.
%p A268343 N:= 1000: # to get all terms <= N
%p A268343 pr:= select(isprime, [$2 .. nextprime(nextprime(N))]):
%p A268343 Np:= nops(pr):
%p A268343 ishermit:= Vector(Np,1):
%p A268343 d:= pr[3..Np] + pr[1..Np-2] - 2*pr[2..Np-1]:
%p A268343 for i from 1 to Np-2 do
%p A268343   if d[i] > 0 then ishermit[i]:= 0
%p A268343 elif d[i] < 0 then ishermit[i+2]:= 0
%p A268343 else ishermit[i]:= 0; ishermit[i+2]:= 0
%p A268343 fi
%p A268343 od:
%p A268343 subs(0=NULL, zip(`*`, pr[1..Np-2],convert(ishermit,list))); # _Robert Israel_, Mar 09 2016
%t A268343 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 *)
%o A268343 (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
%o A268343 (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
%Y A268343 Cf. A269734 (number of hermit primes <= prime(n)).
%K A268343 easy,nonn
%O A268343 1,1
%A A268343 _Karl W. Heuer_, Feb 02 2016
%E A268343 Deleted my incorrect conjecture about asymptotic behavior. - _N. J. A. Sloane_, Mar 10 2016