A120937 -id:A120937 - cp's OEIS frontend

A023186 Lonely (or isolated) primes: increasing distance to nearest prime.

2, 5, 23, 53, 211, 1847, 2179, 3967, 16033, 24281, 38501, 58831, 203713, 206699, 413353, 1272749, 2198981, 5102953, 10938023, 12623189, 72546283, 142414669, 162821917, 163710121, 325737821, 1131241763, 1791752797, 3173306951, 4841337887, 6021542119, 6807940367, 7174208683, 8835528511, 11179888193, 15318488291, 26329105043, 31587561361, 45241670743

Offset: 1

David W. Wilson

Erdős and Suranyi call these reclusive primes and prove that there are an infinite number of them. They define these primes to be between two primes. Hence their first term would be 3 instead of 2. Record values in A120937. - T. D. Noe, Jul 21 2006

			The nearest prime to 23 is 4 units away, larger than any previous prime, so 23 is in the sequence.
The prime a(4) = A120937(3) = 53 is at distance 2*3 = 6 from its neighbors {47, 59}. The prime a(5) = A120937(4) = A120937(5) = A120937(6) = 211 is at distance 2*6 = 12 from its neighbors {199, 223}. Sequence A120937 requires the terms to have 2 neighbors, therefore its first term is 3 and not 2. - _M. F. Hasler_, Dec 28 2015

Paul Erdős and Janos Suranyi, Topics in the theory of numbers, Springer, 2003.

Dmitry Petukhov, Table of n, a(n) for n = 1..56 (first 40 terms from Ken Takusagawa, terms 41..52 from Giovanni Resta)

Related sequences: A023186, A023187, A023188, A046929, A046930, A046931, A051650, A051652, A051697-A051702, A051728, A051729, A051730, A102723.

The distances are in A023187.

p = 0; q = 2; i = 0; Do[r = NextPrime[q]; m = Min[r - q, q - p]; If[m > i, Print[q]; i = m]; p = q; q = r, {n, 1, 152382000}]
Join[{2},DeleteDuplicates[{#[[2]],Min[Differences[#]]}&/@Partition[Prime[ Range[ 2,10^6]],3,1],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]]] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Aug 31 2023 *)

More terms from Jud McCranie, Jun 16 2000

More terms from T. D. Noe, Jul 21 2006

A330428 Smallest prime p such that both nearest primes up and down are farther away than n*log(p).

Original entry on oeis.org

5, 211, 38501, 413353, 10938023, 142414669, 163710121, 8835528511, 31587561361, 343834606051, 1480975873513, 26923643849953

Offset: 1

Steven M. Altschuld, Dec 14 2019

For these numbers, the name "Lowest Frogger Primes" LFP(n) is suggested because (frog) jumps with a length greater than n times the local average are required to bridge the gaps (logs).

Cf. A288908 (with 1*log(P)), A330426 (with 2*log(P)), A330427 (with 3*log(P)).

Cf. A023186, A120937.

{my(npp=2,np=3,n=1);forprime(p=5,10^9,my(d=log(p)*n);if(np-npp>d&&p-np>d,print(np,", ");n++);npp=np;np=p)} \\ Hugo Pfoertner, Dec 14 2019

a(5)-a(9) from Hugo Pfoertner, Dec 14 2019
a(10) from Hugo Pfoertner, Dec 16 2019
a(11)-a(12) from Giovanni Resta, Dec 19 2019

A289154 Smallest prime p > 2^n such that none of p -+ 2^0, p -+ 2^1, p -+ 2^2, ..., p -+ 2^n are prime.

Original entry on oeis.org

5, 23, 53, 211, 251, 787, 787, 1409, 1777, 1777, 1973, 3181, 4889, 8363, 19583, 34171, 66683, 131701, 263227, 527099, 1049011, 2098027, 4196407, 8389001, 16779001, 33555517, 67108913, 134219273, 268435537, 536871743, 1073743303, 2147485673, 4294968857

Offset: 0

Juri-Stepan Gerasimov, Jun 26 2017

nonn

			a(0) = 5 because prime 5 > 2^0 = 1 and none of 5 - 2^0 = 4, 5 + 2^0 = 6 are prime,
a(1) = 23 because prime 23 > 2^1 = 2 and none of 23 - 2^2 = 22, 23 + 2^0 = 24, 23 - 2^1 = 21, 23 + 2^1 = 25 are prime,
a(2) = 53 because prime 53 > 2^2 = 4 and none of 53 - 2^0 = 52, 53 + 2^0 = 54, 53 - 2^1 = 51, 53 + 2^1 = 55, 53 - 2^2 = 49, 53 + 2^2 = 57 are prime.

Charles R Greathouse IV, Table of n, a(n) for n = 0..200

Cf. A120937.

Table[p = NextPrime[2^n]; While[AnyTrue[p + Flatten@ Map[2^Range[0, n] # &, {-1, 1}], PrimeQ], p = NextPrime@ p]; p, {n, 0, 32}] (* Michael De Vlieger, Jun 27 2017 *)

a(n)=if(n<1, return(5)); forprime(p=2^n+1,, for(k=1,n, if(isprime(p+2^k) || isprime(p-2^k), next(2))); return(p)) \\ Charles R Greathouse IV, Jul 07 2017

More terms from Michael De Vlieger, Jun 27 2017

a(15) corrected by Charles R Greathouse IV, Jul 07 2017

cp's OEIS Frontend

A023186 Lonely (or isolated) primes: increasing distance to nearest prime.

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A330428 Smallest prime p such that both nearest primes up and down are farther away than n*log(p).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

A289154 Smallest prime p > 2^n such that none of p -+ 2^0, p -+ 2^1, p -+ 2^2, ..., p -+ 2^n are prime.

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Mathematica

PARI

Extensions