A330428 -id:A330428 - cp's OEIS frontend

A288908 Primes p whose distance from next prime and from previous prime is greater than log(p).

5, 7, 23, 37, 47, 53, 89, 157, 173, 211, 251, 257, 263, 293, 331, 337, 359, 367, 373, 389, 409, 479, 631, 691, 701, 709, 719, 787, 797, 839, 919, 929, 1163, 1171, 1201, 1249, 1259, 1381, 1399, 1409, 1471, 1511, 1523, 1531, 1637, 1709, 1733, 1801, 1811, 1823

Offset: 1

Giuseppe Coppoletta, Jun 19 2017

nonn

Primes preceded and followed by larger-than-average prime gaps (see link), then included in A082885.

			n = 5 is a term because 3 + log(5) < 5 < 7 - log(5).
n = 11 is not a term because 13 - 11 < log(11) = 2.39...

Eric Weisstein's MathWorld, Prime Number Theorem

Cf. A082885, A151799, A151800, A288907, A330426, A330427, A330428.

f:=func;  [p:p in PrimesInInterval(3,2000)|f(p)]; // Marius A. Burtea, Dec 19 2019

Select[Prime@ Range[2, 300], Min@ Abs[# - NextPrime[#, {-1,1}]] > Log[#] &] (* Giovanni Resta, Jun 19 2017 *)

[n for n in prime_range(3,2000) if next_prime(n)-n>log(n) and n-previous_prime(n)>log(n)]

A151799(a(n)) + log(a(n)) < a(n) < A151800(a(n)) - log(a(n)).

A330426 Primes P where the distance to the nearest prime is greater than 2*log(P).

Original entry on oeis.org

211, 2179, 2503, 3967, 4177, 7369, 7393, 11027, 11657, 14107, 16033, 16787, 18013, 18617, 18637, 18839, 19661, 21247, 23719, 24281, 29101, 32749, 33247, 33679, 33997, 37747, 38501, 40063, 40387, 42533, 42611, 44417, 46957, 51109, 51383, 53479, 54217, 55291, 55763

Offset: 1

Steven M. Altschuld, Dec 14 2019

nonn

The author suggests that these numbers be called Double Frogger Primes because two times the distance as the average distance to the nearest neighbor (the log) has to be hopped.

			P = 211 is a term because 199 + 2*log(211) < 211 < 223 - 2*log(211).
P = 199 is not a term because 197 + 2*log(199) > 199.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A288908 (with 1*log(P)), A330427 (with 3*log(P)), A330428.

f:=func;  [p:p in PrimesUpTo(56000)|f(p)];// Marius A. Burtea, Dec 18 2019

q:= 3: state:= false: count:= 0: Res:= NULL:
while count < 100 do
  p:= nextprime(q);
  newstate:= is(p-q > 2*log(q));
  if state and newstate then
    count:= count+1; Res:= Res, q;
  fi;
  q:= p; state:= newstate;
od:
Res; # Robert Israel, Dec 18 2019

lst={};Do[a=Prime[n];If[Min[Abs[a-NextPrime[a,{-1,1}]]]>2*Log[a],AppendTo[lst,a]],{n,1,10000}];lst (* Metin Sariyar, Dec 15 2019 *)
(* Second program: *)
Select[Prime@ Range[10^4], Min@ Abs[# - NextPrime[#, {-1, 1}]] > 2 Log[#] &] (* Michael De Vlieger, Dec 15 2019 *)

More terms from Metin Sariyar, Dec 15 2019

A330427 Primes P where the nearest prime is greater than 3*log(P) away.

Original entry on oeis.org

38501, 58831, 153191, 203713, 206699, 232259, 247141, 250543, 268343, 279269, 286927, 302053, 330509, 362521, 362801, 404597, 413353, 421559, 430193, 438091, 479081, 479701, 485263, 504727, 512207, 515041, 539573, 539993, 546781, 569369, 574859, 590489, 624917

Offset: 1

Steven M. Altschuld, Dec 14 2019

nonn

The author suggests that these numbers be called Triple Frogger Primes because three times the distance as the average distance to the nearest neighbor (the log) has to be hopped.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A288908 (with 1*log(P)), A330426 (with 2*log(P)), A330428 (Lowest Frogger Primes).

f:=func;  [p:p in PrimesUpTo(630000)|f(p)];// Marius A. Burtea, Dec 18 2019

q:= 3: state:= false: count:= 0: Res:= NULL:
while count < 100 do
  p:= nextprime(q);
  newstate:= is(p-q > 3*log(q));
  if state and newstate then
    count:= count+1; Res:= Res, q;
  fi;
  q:= p; state:= newstate;
od:
Res; # Robert Israel, Dec 18 2019

Select[Prime@ Range[10^5], Min@ Abs[# - NextPrime[#, {-1, 1}]] > 3 Log[#] &] (* Michael De Vlieger, Dec 15 2019 *)

lista(nn) = {my(x=2, y=3); forprime(p=5, nn, if(min(p-y, y-x)>3*log(y), print1(y, ", ")); x=y; y=p); } \\ Jinyuan Wang, Mar 03 2020

More terms from Michael De Vlieger, Dec 15 2019

A120937 Least prime such that the distance to the two adjacent primes is 2n or greater.

Original entry on oeis.org

3, 5, 23, 53, 211, 211, 211, 1847, 2179, 2179, 3967, 16033, 16033, 24281, 24281, 24281, 38501, 38501, 38501, 38501, 38501, 58831, 203713, 206699, 206699, 413353, 413353, 413353, 1272749, 1272749, 1272749, 1272749, 2198981, 2198981, 2198981

Offset: 0

T. D. Noe, Jul 21 2006

nonn

Erdos and Suranyi call these reclusive primes and prove that such a prime exists for all n. Except for a(0), the record values are in A023186.

			a(3)=53 because the adjacent primes 47 and 59 are at distance 6 and all smaller primes have a closer distance.

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

Cf. A023186, A087770, A330428.

k=2; Table[While[Prime[k]-Prime[k-1]<2n || Prime[k+1]-Prime[k]<2n, k++ ]; Prime[k], {n,0,40}]

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A288908 Primes p whose distance from next prime and from previous prime is greater than log(p).

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A330426 Primes P where the distance to the nearest prime is greater than 2*log(P).

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A330427 Primes P where the nearest prime is greater than 3*log(P) away.

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A120937 Least prime such that the distance to the two adjacent primes is 2n or greater.

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