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 A268343 Hermit primes: primes which are not a nearest neighbor of another prime. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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-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 A057876 A244282 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

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Last modified November 28 16:34 EST 2021. Contains 349413 sequences. (Running on oeis4.)