A087770 "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.

2, 3, 7, 23, 89, 211, 1847, 2179, 14107, 33247, 38501, 58831, 268343, 1272749, 2198981, 10938023, 72546283, 162821917, 325737821, 2888688863, 6613941601, 11179888193, 24016237123, 96155166493, 179474021633, 215686840471, 633880576177, 1480975873513, 9156364643509

Offset

1,1

Comments

The concept of "lonely prime" is similar to that of maximal prime gaps since lonely primes are increasingly distant from each other.

See A023186 for another version of this sequence, which only requires increasing the minimum of the two gaps to the neighbors. The definition from A023186 seems to be the more common variant. - Hugo Pfoertner, Dec 17 2019

Links

Table of n, a(n) for n=1..29.

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.

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}] (* Robert G. Wilson v *)

Crossrefs

Cf. A000230, A002386, A023186.

Sequence in context: A000230 A256454 A133429 * A237359 A087164 A077213

Adjacent sequences: A087767 A087768 A087769 * A087771 A087772 A087773

Keyword

nonn

Author

Walter Carlini, Oct 03 2003

Extensions

Corrected and extended by Ray Chandler, Oct 06 2003

Offset changed and a(21)-a(27) from Hugo Pfoertner, Dec 17 2019

a(28)-a(29) from Giovanni Resta, Dec 17 2019

Status

approved