A096265 Aloof primes: Total distance between prime and neighboring primes sets record.

2, 3, 5, 7, 23, 53, 89, 113, 211, 1129, 1327, 2179, 2503, 5623, 9587, 14107, 19609, 19661, 31397, 31469, 38501, 58831, 155921, 360749, 370261, 396833, 1357201, 1561919, 4652353, 8917523, 20831323, 38089277, 70396393, 72546283, 102765683

Offset: 1

Author

Rick L. Shepherd, Jun 21 2004

Keywords

• nonn

Examples

			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.
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..55, terms 1..50 from Ken Takusagawa.

Crossrefs

Cf. A031132 (record distances corresponding to a(2) onward), A023186 (lonely primes), A087770 (lonely primes, another definition).

Programs

Mathematica

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 *)
 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 *)

PARI

/* 436272953 is the next-to-the-largest precalculated prime */
/* with which PARI/GP (Version 2.0.17 (beta) at least) can be started */
/* A different program would be required to go beyond a(37)=325737821 */
{r=0; print1("2,"); forprime(p=3,436272953,
s=nextprime(p+1)-precprime(p-1); if(s>r, print1(p,","); r=s))}