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A096265
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Aloof primes: Total distance between prime and neighboring primes sets record.
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2
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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))}
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CROSSREFS
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Cf. A031132 (record distances corresponding to a(2) onward), A023186 (lonely primes), A087770 (lonely primes, another definition).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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