A330427 -id:A330427 - 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

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

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

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