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%I #25 Jul 05 2022 17:22:31
%S 2,3,5,7,23,53,89,113,211,1129,1327,2179,2503,5623,9587,14107,19609,
%T 19661,31397,31469,38501,58831,155921,360749,370261,396833,1357201,
%U 1561919,4652353,8917523,20831323,38089277,70396393,72546283,102765683
%N Aloof primes: Total distance between prime and neighboring primes sets record.
%H Hugo Pfoertner, <a href="/A096265/b096265.txt">Table of n, a(n) for n = 1..55</a>, terms 1..50 from Ken Takusagawa.
%e a(1) = 2 as 2 has only one prime neighbor, 3 and 3-2 = 1, the first possible record. a(2) = 3 because the sum of the distances (gaps) from 3 to its two neighboring primes is 3-2 + 5-3 = 3 > 1, beating the previous record. a(5) = 23 because 23, with 29-19 = 10, is the smallest prime beating a(4) = 7's 11-5 = 6.
%t PrimeNextDelta[n_]:=(Do[If[PrimeQ[n+k], a=n+k; d=a-n; Break[]], {k, 9!}]; d); PrimePrevDelta[n_]:=(Do[If[PrimeQ[n-k], a=n-k; d=n-a; Break[]], {k, n}]; d); q=0; lst={2}; Do[p=Prime[n]; d1=PrimeNextDelta[p]; d2=PrimePrevDelta[p]; d=d1+d2; If[d>q, AppendTo[lst, p]; q=d], {n, 2, 10^4}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Aug 07 2008 *)
%t Join[{2},DeleteDuplicates[{#[[2]],#[[3]]-#[[1]]}&/@Partition[Prime[Range[6 10^6]],3,1],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]]] (* _Harvey P. Dale_, Jul 05 2022 *)
%o (PARI) /* 436272953 is the next-to-the-largest precalculated prime */
%o /* with which PARI/GP (Version 2.0.17 (beta) at least) can be started */
%o /* A different program would be required to go beyond a(37)=325737821 */
%o {r=0; print1("2,"); forprime(p=3,436272953,
%o s=nextprime(p+1)-precprime(p-1); if(s>r, print1(p,","); r=s))}
%Y Cf. A031132 (record distances corresponding to a(2) onward), A023186 (lonely primes), A087770 (lonely primes, another definition).
%K nonn
%O 1,1
%A _Rick L. Shepherd_, Jun 21 2004
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