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A087770
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"Lonely primes": those primes that are locally maximally isolated from the nearest other primes. The differences between each lonely prime and the immediately preceding prime and following primes are both greater than the corresponding differences for all lonely primes earlier in the sequence.
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1
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2, 3, 7, 23, 89, 211, 1847, 2179, 14107, 33247, 38501, 58831, 268343, 1272749, 2198981, 10938023, 72546283, 162821917, 325737821, 2888688863
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| The concept of "lonely prime" is similar to that of maximal prime gaps since lonely primes are increasingly distant from each other.
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EXAMPLE
| a(0) = 2
a(1) = 3 because 3 - 2 = 1 and 5 - 3 = 2
a(2) = 7 because 7 - 5 = 2 (and 2 > 3 - 2) and 11 - 7 = 4 (and 4 > 5 - 3)
a(3) = 23 because 23 - 19 = 4 ( 23 - 19 > 7 - 5) and 29 - 23 = 6 (29 - 23 > 11 - 7)
a(4) = 89 because 89 - 83 = 6 > 23 - 19 and 97 - 89 = 8 > 29 - 23
Note, for example, that 53 is not a lonely prime because 53 - 47 = 6, which is > 23 - 19 however 59 - 53 = 6, which is not > 29 - 23.
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MATHEMATICA
| NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 2; q = 2; r = 3; d = e = 0; Do[ While[ q - p <= d || r - q <= e, p = q; q = r; r = NextPrim[r]]; Print[q]; d = Max[q - p, d]; e = Max[r - q, e]; p = q; q = r; r = NextPrim[r], {n, 1, 40}] (from Robert G. Wilson v)
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CROSSREFS
| Cf. A000230, A002386.
Sequence in context: A002386 A000230 A133429 * A087164 A077213 A112601
Adjacent sequences: A087767 A087768 A087769 * A087771 A087772 A087773
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KEYWORD
| nonn
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AUTHOR
| Walter G. Carlini (541carlini(AT)charter.net), Oct 03 2003
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EXTENSIONS
| Corrected and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Oct 06 2003
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