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

1,1

Links

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

Example

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.

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 *)

Prog

(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))}

Crossrefs

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

Sequence in context: A068710 A120805 A177119 * A356271 A056041 A083017

Adjacent sequences: A096262 A096263 A096264 * A096266 A096267 A096268

Keyword

nonn

Author

Rick L. Shepherd, Jun 21 2004

Status

approved